# Help understanding "Heat capacity"

• nemanjavuk

#### nemanjavuk

Homework Statement:: First of, this is not a "homework" per say, since this is not in my curriculum. If you still want to help, see the description :)
Relevant Equations:: C = Q/(delta T), where delta T is the raise in temperature in Kelvin and Q is the added heat energy

I want to learn about heat capacity. What I know, is that heat capacity is the relationship between added heat and the raise/ fall in temperature.
See "Relevant Equations" for the equation.

My understanding of heat capacity is - without repeating myself again - is, that different objects have different heat capacity. As an example, if you rub the palms of your hands (assuming you have two pair of hands), you will more quickly get your hands hot than, let's say, a metal rod.
So, with this basic out of the way, here comes the very "silly" question. How come, our "hands" are getting warmer more quickly than the metal rod?

That "heat" I am experiencing when rubbing my hands, if we joyfully imagine reaches 100 degrees celsius, would that mean, that it would take less time for my hands to reach 100 degrees than a metal rod? Furthermore, with a more "normal" experiment. If I placed both my hand and a metal rod in 100 degrees celsius boiling water, would my hands reach 100 degrees celsius faster than the metal rod? Evenmore, does this also mean, that we need more Joule to heat up 1 degree of celsius?

This is the fundamental question I am so eager to be enlightened about. I therefor hope I have the "correct" way of understanding heat capacity. Furthermore, sorry in advance if I am missing some physics terms.

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Homework Statement:: First of, this is not a "homework" per say, since this is not in my curriculum. If you still want to help, see the description :)
Relevant Equations:: C = Q/(delta T), where delta T is the raise in temperature in Kelvin and Q is the added heat energy

I want to learn about heat capacity. What I know, is that heat capacity is the relationship between added heat and the raise/ fall in temperature.
See "Relevant Equations" for the equation.

My understanding of heat capacity is - without repeating myself again - is, that different objects have different heat capacity. As an example, if you rub the palms of your hands (assuming you have two pair of hands), you will more quickly get your hands hot than, let's say, a metal rod.
So, with this basic out of the way, here comes the very "silly" question. How come, our "hands" are getting warmer more quickly than the metal rod?

That "heat" I am experiencing when rubbing my hands, if we joyfully imagine reaches 100 degrees celsius, would that mean, that it would take less time for my hands to reach 100 degrees than a metal rod? Furthermore, with a more "normal" experiment. If I placed both my hand and a metal rod in 100 degrees celsius boiling water, would my hands reach 100 degrees faster than the metal rod?

This is the fundamental question I am so eager to be enlightened about. I therefor hope I have the "correct" way of understanding heat capacity. Furthermore, sorry in advance if I am missing some physics terms.
You are conflating many ideas together. Heat capacity is a measure of how much heat you can add to a mass in order to raise its temperature by a degree. Compare this with the analogous idea of the volume capacity of a bucket defined as the amount of water you must add in the bucket it order to raise the level by one inch.

Heat capacity has nothing to do with amount of heat generated per unit time as in rubbing your hand with your other hand as opposed to a metal rod. The analogy here is that the amount of water that you need to add to the bucket to raise its level by one inch will be the same regardless of whether you are adding the water using your kitchen faucet or a fire hose. The difference is that the fire hose will raise the level in less time. Likewise if you use your hands to raise the temperature of a metal rod by one degree it will take longer than if you use a blowtorch.

One more analogy before answering your question. Suppose you are filling a bucket that has a small leak in it that drains some of the water. It will take more time for the level to rise by one inch but the net amount of water that is needed to accomplish that is the same. The analogy holds with heat if you have a mechanism that drains some of the heat that is added. Have you ever wondered why, when you go to the bathroom barefooted, the tile floor feels colder than the little rug in front of the sink? How could that be if both are at room temperature? That's because the thermal conductivity of the tiles is higher than that of the little rug. This is another of saying that the floor sucks heat away from your footies faster than the little rug.

Thermal condctivity is the key idea to answering your question. Hands are bad heat conductors so any heat that is generated remains mostly localized and only a small amount is carried away by perfusion (blood flow.) Metals are good heat conductors so the carry the generated heat much faster than the other hand. Furthermore, metals are generally smooth which means that friction generates less heat per unit time than two hands rubbing together.

So the answer to your question has nothing to do with heat capacity. So to complete the analogy
Rubbing hand with other hand → Lots of water coming in per second and small leak in the bucket
Rubbing hand with metal rod → Little water coming in per second and big leak in the bucket

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• Lnewqban
Cutting to the chase, the formula tells you what you need to know : dump energy into get an increase in temperature ; take energy out (or allow it to leave) to get a decrease.

As well, it's the surface of your hands that are getting warmer. A metal rod has a low [edit: specific] heat capacity, ie: it will warm up much more quickly than something that's mostly water. Of course metal is more conductive so it will also distribute then conduct/convect/radiate heat out faster.

[edit: note I think I've misinterpreted the OP's HC question as SHC, FWIW]

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• Lnewqban
Have you ever wondered why, when you go to the bathroom barefooted, the tile floor feels colder than the little rug in front of the sink?
No. I have floor heating in my bathroom. • • • russ_watters, kuruman and berkeman
You are conflating many ideas together. Heat capacity is a measure of how much heat you can add to a mass in order to raise its temperature by a degree. Compare this with the analogous idea of the volume capacity of a bucket defined as the amount of water you must add in the bucket it order to raise the level by one inch.

Heat capacity has nothing to do with amount of heat generated per unit time as in rubbing your hand with your other hand as opposed to a metal rod. The analogy here is that the amount of water that you need to add to the bucket to raise its level by one inch will be the same regardless of whether you are adding the water using your kitchen faucet or a fire hose. The difference is that the fire hose will raise the level in less time. Likewise if you use your hands to raise the temperature of a metal rod by one degree it will take longer than if you use a blowtorch.

One more analogy before answering your question. Suppose you are filling a bucket that has a small leak in it that drains some of the water. It will take more time for the level to rise by one inch but the net amount of water that is needed to accomplish that is the same. The analogy holds with heat if you have a mechanism that drains some of the heat that is added. Have you ever wondered why, when you go to the bathroom barefooted, the tile floor feels colder than the little rug in front of the sink? How could that be if both are at room temperature? That's because the thermal conductivity of the tiles is higher than that of the little rug. This is another of saying that the floor sucks heat away from your footies faster than the little rug.

Thermal condctivity is the key idea to answering your question. Hands are bad heat conductors so any heat that is generated remains mostly localized and only a small amount is carried away by perfusion (blood flow.) Metals are good heat conductors so the carry the generated heat much faster than the other hand. Furthermore, metals are generally smooth which means that friction generates less heat per unit time than two hands rubbing together.

So the answer to your question has nothing to do with heat capacity. So to complete the analogy
Rubbing hand with other hand → Lots of water coming in per second and small leak in the bucket
Rubbing hand with metal rod → Little water coming in per second and big leak in the bucket
That's just not what textbook says. My textbook says "If the friction heat is applied to a metal rod instead of the skin of your palms (hands) than that rod will not be as heated. We are hear talking about the palms of the hands having a much smaller heat capacity compared to the metal rod".
Again, this is out of my curriculum, I just find heat and energy very fascinating and want to further study it. Hope this is allowed on this site.

Cutting to the chase, the formula tells you what you need to know : dump energy into get an increase in temperature ; take energy out (or allow it to leave) to get a decrease.

As well, it's the surface of your hands that are getting warmer. A metal rod has a low heat capacity, ie: it will warm up much more quickly than something that's mostly water. Of course metal is more conductive so it will also distribute then conduct/convect/radiate heat out faster.
Do you mean "the metal rod will heat up MUCH MORE SLOW compared to the palm of your hands"? And should heat capacity be understood as the amount of energy it takes to heat a surface of an object up/ down by one degree celsius or the amount of energy it takes to heat THE WHOLE object up/ down by one degree celsius?
Sorry in advance if these are very piddly questions.

That's just not what textbook says. My textbook says "If the friction heat is applied to a metal rod instead of the skin of your palms (hands) than that rod will not be as heated. We are hear talking about the palms of the hands having a much smaller heat capacity compared to the metal rod".
Would it be possible for you to take a legible picture of your textbook where it says this and post it? Youmight as well take a picture of the cover because I am curious what textbook says this.

Again, this is out of my curriculum, I just find heat and energy very fascinating and want to further study it. Hope this is allowed on this site.
If you find heat energy fascinating, this site is the correct place to ask your questions and get answers. • Lnewqban
Sure. This is a Danish textbook published by my own professor at DTU which is the university I study at. I used my best English abilities to translate the paragraph for you. Give me 2 min and you will receive the pictures you asked for :-)

• kuruman
Sure. This is a Danish textbook published by my own professor at DTU which is the university I study at. I used my best English abilities to translate the paragraph for you. Give me 2 min and you will receive the pictures you asked for :-)
Have you tried google translate? It does a pretty good job and you can always fix by hand any words that you think are mistranslated.

Would it be possible for you to take a legible picture of your textbook where it says this and post it? Youmight as well take a picture of the cover because I am curious what textbook says this.

If you find heat energy fascinating, this site is the correct place to ask your questions and get answers. Here you go. Here are the pictures requested. If you read from 4.4. (heat capacity), you can see it is written. I think there are some online softwares which can analyse text on pictures if you want it translated :)

#### Attachments

Have you tried google translate? It does a pretty good job and you can always fix by hand any words that you think are mistranslated.
Pretty sure everything I wrote is correctly translated. Only "tilført varmeenergi" I could not find a good English translation, so I went with "added heat energy".

Heat Capacity of an object is simply the change in temperature caused by a change in energy.

For that, the palm of your hand might well heat up faster than a metal rod ; I don't know. Personally, I don't think that's a great example.

Specific Heat Capacity relates to a substance, not a specific object, and uses unary values of molarity, mass or volume.

For that, if for instance you took a piece of metal foil the same mass and areal size of the palm of your hand, it would heat up much faster than the skin of the hand.

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• PeroK
...
Again, this is out of my curriculum, I just find heat and energy very fascinating and want to further study it. Hope this is allowed on this site.
It is very important not to confuse temperature and heat.
https://en.wikipedia.org/wiki/Temperature

https://en.wikipedia.org/wiki/Heat

https://en.wikipedia.org/wiki/Heat_capacity

I respectfully agree, the example of the book could use some improvement: comparing two bodies (copper and aluminum) of similar dimensions should suffice.

For example, the experiment shown in the picture, shows that three bodies of similar mass and different metals, all with similar initial temperature, melt different volumes of wax.
As you can see, the iron sphere accumulated and then liberated more thermal energy than the other two spheres. Last edited:
Here you go. Here are the pictures requested. If you read from 4.4. (heat capacity), you can see it is written. I think there are some online softwares which can analyse text on pictures if you want it translated :)
Thank you for posting the pictures. I translated the figure caption in the first photo:
Danish
Der skal ikke tilføres meget varmeenergitil huden, før dens temperatur stiger meget, sammenlignet med en kraftig metalstang.
(Googletranslate said that "dens" above should be "dets". Is that accurate?)

English
Not much heat energy needs to be applied to the skin before its temperature rises a lot, compared to a strong metal rod.

Since you are interested in learning about this, let's do a sample calculation. The heat capacity ##C## is the mass of the object multiplied by the specific heat ##c## which is something that is measured experimentally and depends on the composition of the object. The specific heat of water is 4200 J/(kg °C) and the specific heat of iron is 450 J/(kg °C) and the average specific heat of the human body (which is mostly water) is 3500 J/(kg °C). I looked up these numbers on the web. Now the key equation to use here is $$Q=mc\Delta T$$ where
##Q=~## the heat added to the object
##c=~## the specific heat of the object
##m=~## the mass of the object
##\Delta T=~## the rise in temperature of the object when amount of heat ##Q## is added to it.

Now let's see how much heat is needed to raise by 1 °C the temperature of a hand that has mass 0.5 kg and compare that with an iron rod of the same mass.
##Q_{\text{hand}}=0.5~(\text{kg}) \times 3500~(\frac{\text{J}}{\text{kg}~^o\text{C}})\times1~^o\text{C} = 1750~\text{J}.##
##Q_{\text{rod}}=0.5~(\text{kg}) \times 450~(\frac{\text{J}}{\text{kg}~^o\text{C}})\times1~^o\text{C} = 225~\text{J}.##
It takes 7.8 times more heat to raise the temperature of the hand by 1 degree. This seems to contradict the caption of the figure that I translated. This conclusion follows from the calculations which assume that in both cases the heat that is added is distributed equally over the entire mass of the object and that the temperature is the same everywhere inside each object as heat is added. In reality, as I indicated before, when you rub your hands together the heat stays on your skin and is diffused slowly into the body via the bloodstream. The iron rod, being a better conductor of heat, carries the heat into its interior much more quickly.

I do not understand the context of what the textbook is trying to say in that passage therefore I cannot conclude that the book is wrong. Instead I tried to explain to you what I know is correct.

• PeroK and Lnewqban
Danish
Der skal ikke tilføres meget varmeenergitil huden, før dens temperatur stiger meget, sammenlignet med en kraftig metalstang.
(Googletranslate said that "dens" above should be "dets". Is that accurate?)

English
Not much heat energy needs to be applied to the skin before its temperature rises a lot, compared to a strong metal rod.
So, I am not native in Danish, but I am in Swedish which is fairly similar (at least the written language).

Dens vs dets: Variations of the same word that is conjugated based on the gender of the noun it refers to. In this case "hud" (skin) in definite form "huden" has common gender making "dens" the correct form and Google Translate incorrect - possibly caused by the punctuation.

"Kraftig" has (at least in Swedish) several similar meanings. The meaning here is likely more "thick" than "strong" and drawing the mind towards something pretty heavy - significantly more heavy than a hand - making the comparison of equal mass objects unlikely as the intended comparison. (I'm not saying it is pedagogical, just interpreting what the author might have intended.)

• kuruman
So, I am not native in Danish, but I am in Swedish which is fairly similar (at least the written language).

Dens vs dets: Variations of the same word that is conjugated based on the gender of the noun it refers to. In this case "hud" (skin) in definite form "huden" has common gender making "dens" the correct form and Google Translate incorrect - possibly caused by the punctuation.

"Kraftig" has (at least in Swedish) several similar meanings. The meaning here is likely more "thick" than "strong" and drawing the mind towards something pretty heavy - significantly more heavy than a hand - making the comparison of equal mass objects unlikely as the intended comparison. (I'm not saying it is pedagogical, just interpreting what the author might have intended.)
With your fair understanding of Danish and your excellent understanding of physics pedagogy, what would you say the intended comparison is and what point is its author trying to make? I cannot deduce that based on physics alone.

the text is even confusing on a volumetric basis. The specific heat of steel is close to 1/8 that of water; yet the density is close to 8 times. so the volumetric heat capacity of steel and water are nearly identical. 60 vs 62 Btu/ft3-F (pardon my US engineer accent here).

EDIT steel at 4.02MJ/m3-K vs water at 4.16

• Lnewqban
Thank you for posting the pictures. I translated the figure caption in the first photo:
Danish
Der skal ikke tilføres meget varmeenergitil huden, før dens temperatur stiger meget, sammenlignet med en kraftig metalstang.
(Googletranslate said that "dens" above should be "dets". Is that accurate?)

English
Not much heat energy needs to be applied to the skin before its temperature rises a lot, compared to a strong metal rod.

Since you are interested in learning about this, let's do a sample calculation. The heat capacity ##C## is the mass of the object multiplied by the specific heat ##c## which is something that is measured experimentally and depends on the composition of the object. The specific heat of water is 4200 J/(kg °C) and the specific heat of iron is 450 J/(kg °C) and the average specific heat of the human body (which is mostly water) is 3500 J/(kg °C). I looked up these numbers on the web. Now the key equation to use here is $$Q=mc\Delta T$$ where
##Q=~## the heat added to the object
##c=~## the specific heat of the object
##m=~## the mass of the object
##\Delta T=~## the rise in temperature of the object when amount of heat ##Q## is added to it.

Now let's see how much heat is needed to raise by 1 °C the temperature of a hand that has mass 0.5 kg and compare that with an iron rod of the same mass.
##Q_{\text{hand}}=0.5~(\text{kg}) \times 3500~(\frac{\text{J}}{\text{kg}~^o\text{C}})\times1~^o\text{C} = 1750~\text{J}.##
##Q_{\text{rod}}=0.5~(\text{kg}) \times 450~(\frac{\text{J}}{\text{kg}~^o\text{C}})\times1~^o\text{C} = 225~\text{J}.##
It takes 7.8 times more heat to raise the temperature of the hand by 1 degree. This seems to contradict the caption of the figure that I translated. This conclusion follows from the calculations which assume that in both cases the heat that is added is distributed equally over the entire mass of the object and that the temperature is the same everywhere inside each object as heat is added. In reality, as I indicated before, when you rub your hands together the heat stays on your skin and is diffused slowly into the body via the bloodstream. The iron rod, being a better conductor of heat, carries the heat into its interior much more quickly.

I do not understand the context of what the textbook is trying to say in that passage therefore I cannot conclude that the book is wrong. Instead I tried to explain to you what I know is correct.
Hello again!
As Orodruin said. "dens" is just a word used to refer to what is being said. "dens" is referring to THE HAND(s). I was under the impression that heat capacity is how much joule is needed to make the surface (or the whole) of the hand hot. So it would take fewer joule to make your hands 100 degrees celsius and therefore less time than a metal rod.

In general a "metal" has a lower specific heat capacity than flesh (which is mostly water). "Specific" here means "per kilogram" of material. So a given mass of metal shows temperature rising faster than the same mass of water (given the same heat input).

This is assuming (as noted by previous posts) that the heat is not also leaving the metal or flesh.

Overall, the example of rubbing hands as a heat source raises too many questions and qualifiers (assuming this and assuming that). In my opinion it is a bad example to teach someone about "heat capacity."

• Lnewqban
My textbook just says the opposite. That the surface (the flesh) of the hand has a lower heat capacity than the metal rod.

My textbook just says the opposite. That the surface (the flesh) of the hand has a lower heat capacity than the metal rod.
The book is talking about heat capacity, not specific heat capacity. With a thick metal rod, it supposedly has significantly higher mass than the skin of your palm unlike the equal mass example presented in #14 or the specific heat discussed in #19. While the book may not be directly wrong, the example is not very well chosen and possibly misleading.

• russ_watters
The book is talking about heat capacity, not specific heat capacity. With a thick metal rod, it supposedly has significantly higher mass than the skin of your palm unlike the equal mass example presented in #14 or the specific heat discussed in #19. While the book may not be directly wrong, the example is not very well chosen and possibly misleading.
So, to be 100% clear. Heat Capacity (not specific heat capacity) is the amount of energy (if we measure it in Joule) that is needed to raise the temperature of an object (where object is ANYTHING THAT CAN BE COLDER OR MORE HOT) by 1 degree celsius. Is this ONLY the surface area that we are talking about, or the WHOLE object, like the whole metal rod (both inside and outside) that are being being more hot?

My textbook just says the opposite. That the surface (the flesh) of the hand has a lower heat capacity than the metal rod.
Of course it does. Your book is, in a sense, comparing apples and oranges which, for someone like you who tries to understand these concepts, could be confusing. The issue can be resolved by comparing apples with apples which is where specific heat comes in.

I just noticed that @Orodruin already said what I was going to say, so I will stop my explanation here.

So, with this basic out of the way, here comes the very "silly" question. How come, our "hands" are getting warmer more quickly than the metal rod?
Now I think you understand that "skin" and "hand" have different heat capacities and more or less equal specific heats so when you touch something hot at first the heat goes into your skin and not your hand. Likewise, if you place a metal pot on a hot stove, the heat first goes into the surface of the pot. A layer of metal equal in thickness to a layer of skin of equal mass will heat up more quickly than the skin because it has a lower specific heat and therefore requires fewer Joules of energy for its temperature to rise by one degree.

So, to be 100% clear. Heat Capacity (not specific heat capacity) is the amount of energy (if we measure it in Joule) that is needed to raise the temperature of an object (where object is ANYTHING THAT CAN BE COLDER OR MORE HOT) by 1 degree celsius. Is this ONLY the surface area that we are talking about, or the WHOLE object, like the whole metal rod (both inside and outside) that are being being more hot?
The entire object as a whole. Then the question becomes what you consider an ”object”. The treatment with heat capacity assumes an object in internal equilibrium (so homogeneous temperature). If your heating process is fast and you look at time scales shorter than the typical time for the object to equilibrate, then you need to start solving partial differential equations.

Of course it does. Your book is, in a sense, comparing apples and oranges which, for someone like you who tries to understand these concepts, could be confusing. The issue can be resolved by comparing apples with apples which is where specific heat comes in.

I just noticed that @Orodruin already said what I was going to say, so I will stop my explanation here.

Now I think you understand that "skin" and "hand" have different heat capacities and more or less equal specific heats so when you touch something hot at first the heat goes into your skin and not your hand. Likewise, if you place a metal pot on a hot stove, the heat first goes into the surface of the pot. A layer of metal equal in thickness to a layer of skin of equal mass will heat up more quickly than the skin because it has a lower specific heat and therefore requires fewer Joules of energy for its temperature to rise by one degree.
And just to be completely sure. "requires fewer Joules of energy for its temperature to rise by one degree." are you here talking about ONLY the surface of the pot or the whole pot? Just to be sure if heat capacity is talking only about the surface area of an object or the whole object.

Just to be sure if heat capacity is talking only about the surface area of an object or the whole object.
Whole object. You should be able to see from the equations why the idea makes no sense to say for a surface. It's for an object with mass and volume so in effect a surface has zero capacity to store heat.

• nemanjavuk
Whole object. You should be able to see from the equations why the idea makes no sense to say for a surface. It's for an object with mass and volume so in effect a surface has zero capacity to store heat.
If you look at post #3 made by hmmm27 you can see why I was confused, since he talked about the surface (while my original example was the palm of your hands).
I have one last question regarding heat capacity then. If, let's say, a rod (don't matter what it is made out of) has a heat capacity at around 2.3J/Celsius, would that mean, that it would take 2.3 Joules of energy to raise the whole object of the rod by 1 celsius (If we ignore the lost of heat to the surroundings)? If so, I am a bit curious again. Since, the surface of the rod will always be hotter than than the whole rod?
and by the way, thank you so much for elaborating on my question!

If you look at post #3 made by hmmm27 you can see why I was confused, since he talked about the surface (while my original example was the palm of your hands).
It's a symptom of the poorly chosen example in the book. Usually to start with this concept you talk about uniform heating of an object. When rubbing your hands together you are heating part of an object (you), unevenly. Here, the "surface" of your skin is actually some thin but non-zero thickness of skin.
I have one last question regarding heat capacity then. If, let's say, a rod (don't matter what it is made out of) has a heat capacity at around 2.3J/Celsius, would that mean, that it would take 2.3 Joules of energy to raise the whole object of the rod by 1 celsius (If we ignore the lost of heat to the surroundings)?
Yes.
If so, I am a bit curious again. Since, the surface of the rod will always be hotter than than the whole rod?
In the real world, if you want to get complicated, objects are never completely uniform in temperature. But it's exceptionally difficult to deal with that mathematically and the difference is often small enough to ignore (using simplifying assumptions). However, there are methods to deal with uneven heat transfer, often involving differential equations, as @Orodruin mentioned. The undergrad engineering course "Heat Transfer" is all about this issue and is one of the harder core engineering classes.

Also note: care must be taken when talking about this issue as the definitions used in science aren't quite the same as the colloquial definitions for terms like "heat" (noun or verb) or "hot".
and by the way, thank you so much for elaborating on my question!
You're welcome!

• nemanjavuk
It's a symptom of the poorly chosen example in the book. Usually to start with this concept you talk about uniform heating of an object. When rubbing your hands together you are heating part of an object (you), unevenly. Here, the "surface" of your skin is actually some thin but non-zero thickness of skin.

Yes.

In the real world, if you want to get complicated, objects are never completely uniform in temperature. But it's exceptionally difficult to deal with that mathematically and the difference is often small enough to ignore (using simplifying assumptions). However, there are methods to deal with uneven heat transfer, often involving differential equations, as @Orodruin mentioned. The undergrad engineering course "Heat Transfer" is all about this issue and is one of the harder core engineering classes.

Also note: care must be taken when talking about this issue as the definitions used in science aren't quite the same as the colloquial definitions for terms like "heat" (noun or verb) or "hot".

You're welcome!
You know what, you just answered all ,y questions. Thank you so much. I am relieved to finally understand heat capacity. Thanks once again to you and all the rest who helped me understanding. Have a beautiful week ahead!

• Lnewqban, berkeman and russ_watters