# How much warmer is 10 degree Celsius than 5 degrees Celsius?

Hi everyone- this is my first post on the Forum. I'm a H.S. Mathematics teacher with a reasonable understanding of basic Physics for someone with no degree in the subject (Treat me like a solid H.S. Physics student in terms of my understanding).

So I was in Canada this past week, and I began to wonder, "How much warmer is 10 degrees than 5 degrees?" The obvious (and uninteresting answer) is 5 degrees, but I'm thinking in terms of a rate of change, or a comparison. So one might be tempted to say that it is twice as warm, but this is clearly wrong in terms of the heat (Kinetic Energy) present in the atmosphere.

My next thought was: since absolute zero is around -273 Celcius, that means that at 5°C you have enough heat to raise the temperature 278° (from absolute zero), as compared to 283° at 15°C. That means it's only about (5/278)*100 percent warmer- a rather modest increase.

But then I got to wondering if heat works like that. Is heat a linear representation of the Kinetic Energy present in a system? Or is it logarithmic, exponential, etc.?

In other words: Is the amount of heat need to go from 1°K to 2°K the SAME as the amount of heat required to go from say, 2001°K to 2002°K?

Sorry if this is a long winded question- I just wanted to explain myself as clearly as possible. Thanks in advance for your help- I hope that you find this question interesting. My office-mates and I have had a great time trying to figure out the answer, but for all we know that is a trivial old chestnut of a question in the Physics community

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No, in general the amount of heat to change the temperature by one unit (see specific heat) depends on temperature.
For example, for a solid material, the heat to increase temperature from 1K to 2 K is much less than the heat to increase temperature by 1K around room temperature.

I don't see how Canada is relevant to this. You can ask the same questions in Fahrenheit scale, eh? :)

Borek
Mentor
Amount of heat required to heat something up by 1 deg depends on the specific heat of the something, and as specific heat is a function of temperature amount of heat required to heat something is a function of temperature as well. That's what nasu refers to.

On a less general scale it is not difficult to show examples of (nearly) perfect linearity.

For ideal gases average kinetic energy of molecules is directly proportional to the temperature:

$$E = \frac 3 2 kT$$

(see http://hyperphysics.phy-astr.gsu.edu/HBASE/kinetic/kintem.html#c1 for more details).

For solids around room temperature specific heat is almost constant (compare http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html).

DrDu
Well, another point of view is to consider entropy: ΔS=Q/T where T is absolute temperature and Q is heat exchanged (reversibly). The same amount of heat will lead to a much larger change in entropy at lower temperatures than at higher temperatures. Therefore it is sometimes preferable to consider inverse temperature ##\beta=1/kT##.

sophiecentaur
Gold Member
@Mike
You have grasped the significance of the absolute temperature - which is the most important thing, in terms of Physics. The thermal Energy in a body is related to this temperature scale and not to any other. (The number of times you hear that 20C is 'twice as hot as' 10C !!!)
Any temperature scale that's used, will be designed to be reproducible and easily specified so that measurements can be made to agree from one lab to another. Early scales used 'fixed points' (boiling / freezing / triple point etc) and used the change in volume of a fluid or resistance of a metal coil (and probably other things) to give a measurement. The scales would assume 'equal step sizes' for, say 100 divisions but, of course, if the quantity measured doesn't follow a linear law, the scales can depart from each other between the fixed points. The thermodynamic scale is an attempt to unify all those scales and the Degree C is what it's based on.
The Kelvin scale (which has the same 'step size' as the Celcius scale, by definition) is based on the average KE of the particles. For an ideal gas, the energy needed to raise the temperature of a given mass by 1C is the same, whatever the absolute temperature. A real gas (or liquid or solid) will not behave this way because some of the energy put in may alter the Potential Energy (Van der Waal's forces etc) so the energy transfer isn't so straightforward.

I find the 'absoluteness' of absolute zero an interesting idea. In many ways, it's like c. Although the scale is linear, the scale of difficulty in getting to very low temperatures (as with getting to very high speeds) is far from linear. So you could say the attainability scale is more or less linear at the vast mid range but stretches without limit near 0K. More and more energy is needed to refrigerate an object to nearer and nearer Zero.

I notice you use the term "Warmness", which suggests you are trying to relate all this to the subjective effect (?). You don't tend to find the word 'warm' in Physics books. Its use in everyday life tends to be limited to a narrow range of temperatures, slotted in between 'too damned cold' and 'too damned hot' (about 40°C range). As far as our bodies are concerned, this range of temperatures is much more significant than others as our proteins don't survive well outside this range.

Andrew Mason
Homework Helper
My next thought was: since absolute zero is around -273 Celcius, that means that at 5°C you have enough heat to raise the temperature 278° (from absolute zero), as compared to 283° at 15°C. That means it's only about (5/278)*100 percent warmer- a rather modest increase.

But then I got to wondering if heat works like that. Is heat a linear representation of the Kinetic Energy present in a system? Or is it logarithmic, exponential, etc.?

In other words: Is the amount of heat need to go from 1°K to 2°K the SAME as the amount of heat required to go from say, 2001°K to 2002°K?
The simplest case is an ideal monatomic gas, such as He. The heat capacity of an ideal monatomic gas is not temperature dependent so it takes the same amount of energy to raise a mole of such gas by a one degree at any temperature.

Temperature is a measure of the average translational kinetic energy of the molecules in the gas. So if you double the temperature you double the translational kinetic energy of the gas molecules.

If you want to look at the gases in the atmosphere it gets a bit more complicated but for normal temperature ranges, the atmosphere approximates an ideal gas.

Humans are sensitive to the rate at which our bodies lose heat. Our bodies are sensitive to temperatures in the 0-45 C range. The human body is trying to maintain a temperature of 37C. At a normal metabolic rate the body the body keeps its 37C temperature nicely at about 20C. Higher temperatures mean the body has to find a way to cool itself (sweat = cooling by evaporation). At lower temperatures, we have to find ways of preventing heat loss (clothing).

So, humans sense temperature in terms of the effect it has on our bodies ability to maintain its core temperature at 37C, which is not in proportion to the absolute temperature.

AM

Dale
Mentor
So I was in Canada this past week, and I began to wonder, "How much warmer is 10 degrees than 5 degrees?" The obvious (and uninteresting answer) is 5 degrees, but I'm thinking in terms of a rate of change, or a comparison. So one might be tempted to say that it is twice as warm, but this is clearly wrong in terms of the heat (Kinetic Energy) present in the atmosphere.

My next thought was: since absolute zero is around -273 Celcius, that means that at 5°C you have enough heat to raise the temperature 278° (from absolute zero), as compared to 283° at 15°C. That means it's only about (5/278)*100 percent warmer- a rather modest increase.

But then I got to wondering if heat works like that. Is heat a linear representation of the Kinetic Energy present in a system? Or is it logarithmic, exponential, etc.?

In other words: Is the amount of heat need to go from 1°K to 2°K the SAME as the amount of heat required to go from say, 2001°K to 2002°K?
I think that the only answer is "it depends". It depends on exactly how you want to do this comparison. The "uninteresting" calculation you did is essentially correct for the thermal energy contained an ideal gas, which is kind of the "default" scenario to consider. With other materials the calculation for the thermal energy would be different.

However, I like Andrew Mason's point about talking about the rate of heat transfer from a human body. This would allow all sorts of interesting physical principles to be illustrated. You could compare differences in radiative heat transfer between 5º C and 10º C (proportional to the difference in the 4th power of the temperature). You could compare differences in convective heat transfer. You could compare differences in conductive heat transfer assuming different levels of clothing (proportional to the temperature difference). You could compare rate of heat lost by 10º C still air to 10º C still water.

I think it's worth mentioning that change in temperature behaves differently around phase change temperatures. For example, boiling liquid water will not increase in temperature if you apply excess energy. The steam does, of course, but not the liquid.

sophiecentaur
Gold Member
Well, another point of view is to consider entropy: ΔS=Q/T where T is absolute temperature and Q is heat exchanged (reversibly). The same amount of heat will lead to a much larger change in entropy at lower temperatures than at higher temperatures. Therefore it is sometimes preferable to consider inverse temperature ##\beta=1/kT##.
A good way to look at it.

Wow- thanks so much everyone for all the detailed replies. This has been truly eye-opening. I had theorized that it might take more energy to heat a substance the "next degree" as the starting temp went up, so it's cool to see that intuition was more or less on target.

If I'm understanding everyone, specific heat is a function of temperature, but over the range of temperatures that us humans are comfortable in, the specific heat would be relatively constant. For an ideal gas the specific heat is independent of temperature ( I assume this is only at temperatures that maintain the gas in a gaseous state- is that correct? Or does the specific heat remain stable even as a solid?), and someone mentioned the atmosphere we breathe is pretty close to an ideal gas in practice.

So I guess I am now wondering... Is there a sensible way to answer my question?

If I refine the question to ask, how much more energy is required to heat the atmosphere to 10°C, as opposed to heating the atmosphere to 5°C, is it possible to come up with a sensible answer?

@nasu - The reason Canada was significant is that what got me thinking about this was realizing that the increase in heat to go from 5 to 10 F could not be the same (at least as a percentage) as the increase in heat to go from 5 to 10 C.

@sophiecentaur - This really made me laugh and smile:
"Its use in everyday life tends to be limited to a narrow range of temperatures, slotted in between 'too damned cold' and 'too damned hot' (about 40°C range)."

If I refine the question to ask, how much more energy is required to heat the atmosphere to 10°C, as opposed to heating the atmosphere to 5°C, is it possible to come up with a sensible answer?
What is the initial condition of the atmosphere?

Borek
Mentor
For an ideal gas the specific heat is independent of temperature ( I assume this is only at temperatures that maintain the gas in a gaseous state- is that correct? Or does the specific heat remain stable even as a solid?)
Ideal gas never condenses nor solidifies.

The reason Canada was significant is that what got me thinking about this was realizing that the increase in heat to go from 5 to 10 F could not be the same (at least as a percentage) as the increase in heat to go from 5 to 10 C.
While these are the same numbers of degrees, 1°F does not equal 1°C (in terms of difference, not the temperature itself). And it is not because these are separate scales, Kelvin and Celsius are separate scales too, but 1K = 1°C.

In a way it is like measuring distance in feet and meters. K and °C both "use a meter", they just measure the distance from different starting points, F "uses feet" (and even another starting point).

sophiecentaur
Gold Member
If I refine the question to ask, how much more energy is required to heat the atmosphere to 10°C, as opposed to heating the atmosphere to 5°C, is it possible to come up with a sensible answer?

@nasu - The reason Canada was significant is that what got me thinking about this was realizing that the increase in heat to go from 5 to 10 F could not be the same (at least as a percentage) as the increase in heat to go from 5 to 10 C.
The 'step size' is different in the Farenheit scale from the Celcius scale so, of course 5 to 10 is a different change in absolute temperature. The ratio is about 5:9.

When you say "atmosphere", a lot could depend upon humidity and whether there are water droplets (mist) . The energy involved in vaporising and condensing is considerable(so-called Latent Heat of Vaporisation) and that could make a big difference. Then there's the pressure / density.
Is this question based on a practical application - e.g. space heating? If it is, then there a number of parameters that need to be considered and your question needs to include more details.

It is important to realize that temperature is a property of some material sample, and is related to heat energy, but it is not a direct measure of heat energy.

The amount of heat in 1 gram of 25* C water and in 1 gram of 25* C aluminum is different, for example.

Temperature and heat are related, but different measurements.

It is important to realize that temperature is a property of some material sample, and is related to heat energy, but it is not a direct measure of heat energy.

The amount of heat in 1 gram of 25* C water and in 1 gram of 25* C aluminum is different, for example.

Temperature and heat are related, but different measurements.

Thank you. I have been slowly coming to this conclusion as I read through the replies above.

I guess there just isn't any way to answer the question in general in the way I had hoped. This was fun, and educational. Thanks again to everyone for contributing

Andrew Mason
Homework Helper
@nasu - The reason Canada was significant is that what got me thinking about this was realizing that the increase in heat to go from 5 to 10 F could not be the same (at least as a percentage) as the increase in heat to go from 5 to 10 C.
Some Canadians complained in the 1970s that moving to the Celsius system for temperature would cost us more in energy costs for that very reason - it costs more to raise a temperature by 5°C than 5°F. And by golly they were right. My heating cost in 1970 under fahrenheit was a tiny fraction of what it is now with these Celsius temperatures. Sure there has been inflation, but that's not all of it. I have a friend in the U.S. who heats in Fahrenheit whose heating costs today are way less than mine here in Canada.

sophiecentaur
Gold Member
If your Canadian friends can't detect that sort of difference with their bodies, I guess they must be wearing some very insulating clothing. No wonder those lumberjack shirts are so popular.

I have a friend in the U.S. who heats in Fahrenheit whose heating costs today are way less than mine here in Canada.
Sorry for asking the obvious, but have you compared the other things, such as the cost of fuel, mean external temperature during the cold season, the preferred internal temperatures, and the houses' thermal insulation?

Borek
Mentor
Perhaps its the time to propose a new temperature scale, Dumb degrees, abbreviated °D, defined as 1/200th of the distance between water freezing and boiling. That would half heating costs.

Andrew Mason