How does stirring a cup of tea affect the entropy of the tea and the universe?

• bananabandana
In summary, the conversation discusses a problem involving a mug of tea in thermal contact with its surroundings at different temperatures. The problem also provides information about the density and specific heat capacity of the tea. The conversation goes on to discuss the changes in entropy of the tea and the universe in two different scenarios, and how work is done on the tea by stirring it. The conversation also touches on the first law of thermodynamics and the role of entropy in determining the final equilibrium temperature of the tea.
bananabandana

Homework Statement

A mug containing a volume of ta initially at 90oC is in thermal contact with the enivronment at 20oC.

Density of tea = ##10^{3}kgm^{-3}##
Constant pressure specific heat capacity of tea: ##4.20 \times 10^{3} Jkg^{-1}K^{-1}##

...
b) After reaching equilibrium, 0.01 J of work is done on the tea by stirring it after which it settles into equilibrium again. Calculate the resulting entropy chagne of the tea and of the universe

Homework Equations

[1] $$\Delta S_{Univ} = C_{P} \bigg( \frac{T_{0}-T_{F}}{T_{R}} + ln \frac{T_{I}}{T_{F}} \bigg)$$
##T_{I}## and ##T_{F}## are the inital and final temperatures respectively.

[2] $$\Delta S_{Tea} = C_{P} ln\frac{T_{F}}{T_{I}}$$

The Attempt at a Solution

Let ##T_{eq}## be the equilibrium temperature of the tea (i.e ##293K##/##20^{o}C##)

(As an aside - we've done work on the tea, but it's volume hasn't increased - how does that make sense? Or is it meant to very subtly expand?)

What I'd like to do is to say that ##dU = 0 \implies dQ = \Delta W ## - but that doesn't seem a very sensible statement to make - unless we make some argument that the tea must lose all of the energy it just gained as heat when it returns to equilibrium?

If I do say that, then it follows that

[3] $$C_{P}dT = \Delta W \implies \Delta T = \frac{\Delta W}{C_{P}}$$

From which it is then trivial to get a value for the entropy for tea and universe by substituting into the equations [1] and [2] above...

Really appreciate the help! Thanks.

Do they tell you how much tea there is?

Before you get to part (b), you should focus on the changes in entropy of the system and the surroundings if part (a). So, how much does the entropy of the system change in part (a) and how much does the entropy of the reservoir (environment) change?

You asked about the work in part (b). For your edification, pressure-volume work is not the only kind of work than can be applied to a system. In this case, stirring the tea does work on the liquid by deforming the liquid; the work is associated with viscous stresses that develop as a result of the deformation.

Yes, volume is ##2.5 \times 10^{-4}m^{3}## - I've found the general expression for the change in the entropy of the universe and the tea in eqns (1) and (2),above? [These were the results given to derive, so I assume they're right!] To get a numerical answer, I would just stick in ## T_{I}=90^{0}C## and ##T_{F}=20^{o}C##,right?

(Oh - and yep, that makes sense - but how come we don't consider these in the first law: ## dU = TdS - PdV## - is that due to the implicit assumption of reversibility, which including viscosity won't allow? )

bananabandana said:
Yes, volume is ##2.5 \times 10^{-4}m^{3}## - I've found the general expression for the change in the entropy of the universe and the tea in eqns (1) and (2),above? [These were the results given to derive, so I assume they're right!] To get a numerical answer, I would just stick in ## T_{I}=90^{0}C## and ##T_{F}=20^{o}C##,right?

(Oh - and yep, that makes sense - but how come we don't consider these in the first law: ## dU = TdS - PdV## - is that due to the implicit assumption of reversibility, which including viscosity won't allow? )
The first law does apply to this situation. For the system in this problem, ##\Delta U = Q-W##. The equation you wrote, ## dU = TdS - PdV## is not the first law. It just tells us the relationship between the changes in U, S, and V between two closely neighboring (i.e., differentially separated) thermodynamic equilibrium states.

In part (b) of your problem, the initial temperature of the tea is 20 C. What is the final equilibrium temperature of the tea (that is in contact with a reservoir at 20 C)? What is the change in entropy of the tea in part (b), given that entropy is a physical property of the tea that is a function only of its initial and final states?

Chestermiller said:
The first law does apply to this situation. For the system in this problem, ##\Delta U = Q-W##. The equation you wrote, ## dU = TdS - PdV## is not the first law. It just tells us the relationship between the changes in U, S, and V between two closely neighboring (i.e., differentially separated) thermodynamic equilibrium states.

In part (b) of your problem, the initial temperature of the tea is 20 C. What is the final equilibrium temperature of the tea (that is in contact with a reservoir at 20 C)? What is the change in entropy of the tea in part (b), given that entropy is a physical property of the tea that is a function only of its initial and final states?

Ah, I see.

I'm guessing you're wanting me to say the ##dS=0## - since the temperature of the tea at the end will be the same as the temperature at the beginning. I guess that makes sense for the tea - like I said earlier, it seems sensible that the tea loses all the extra internal energy it gains. But the surroundings will have to take in that heat. But assuming that the surroundings are a large heat bath, that difference is negligible, so- do we say..
$$\Delta S_{Tea} = 0, \Delta S_{Surround} \approx 0$$
$$\Delta S_{Univ} = 0$$

Though that does seem to me a bit odd...

bananabandana said:
Ah, I see.

I'm guessing you're wanting me to say the ##dS=0## - since the temperature of the tea at the end will be the same as the temperature at the beginning. I guess that makes sense for the tea - like I said earlier, it makes sense that the tea loses all the extra internal energy it gains. But the surroundings will have to take in that heat. But assuming that the surroundings are a large heat bath, that difference is negligible, so- do we say..
$$\Delta S_{Tea} = 0, \Delta S_{Surround} \approx 0$$
$$\Delta S_{Univ} = 0$$

Though that does seem to me a bit odd...

Also - regarding the point about entropy and state variables. Is there an intuitive justification of the fact that any given state variable ##S,U##etc., can be written explicitly in terms of only two variables i.e ## S=S(T,P)=S(V,N)## or do I need to start doing some more theoretical thermodynamics/stat physics to understand this? e.g phase space, Liouville..

bananabandana said:
Ah, I see.

I'm guessing you're wanting me to say the ##dS=0## - since the temperature of the tea at the end will be the same as the temperature at the beginning. I guess that makes sense for the tea - like I said earlier, it seems sensible that the tea loses all the extra internal energy it gains. But the surroundings will have to take in that heat. But assuming that the surroundings are a large heat bath, that difference is negligible, so- do we say..
$$\Delta S_{Tea} = 0, \Delta S_{Surround} \approx 0$$
$$\Delta S_{Univ} = 0$$

Though that does seem to me a bit odd...
You said that 0.01 J of work was done in stirring the tea (this seems awfully low. Did you mean 0.01 kJ?). Since the internal energy of the tea didn't change, where did that work go?

That's what they put in the question! I'm assuming that the work went out as heat into the surroundings when the tea cooled?

bananabandana said:
Also - regarding the point about entropy and state variables. Is there an intuitive justification of the fact that any given state variable ##S,U##etc., can be written explicitly in terms of only two variables i.e ## S=S(T,P)=S(V,N)## or do I need to start doing some more theoretical thermodynamics/stat physics to understand this? e.g phase space, Liouville..
Is the phase rule at a sufficiently fundamental level for you? What are the degrees of freedom for a single phase substance of constant composition?

Or, for a single phase substance, if entropy per unit mass is a physical property, it can depend on T, P, and V, where V is the specific volume. But, the equation of state eliminates one of these. Does that work for you?

bananabandana said:
That's what they put in the question! I'm assuming that the work went out as heat into the surroundings when the tea cooled?
This is correct. So, what is the change in entropy of the surroundings?

The change in entropy of the surroundings is zero? Like I said above? Or, from equation (1) since the surroundings are a heat bath, there is no change in temperature, therefore no change in entropy...?

Phase rule makes sense, had to look it up -## F= C-P+2## - in this case, ## C=1## (assuming we can model tea as pure water) ##P=1, \therefore F=2##

bananabandana said:
The change in entropy of the surroundings is zero? Like I said above? Or, from equation (1) since the surroundings are a heat bath, there is no change in temperature, therefore no change in entropy...?
The change in entropy of the heat bath is 0.01/293 J/K. This is the amount that the bath receives (ideally reversibly) from the tea. So the entropy is generated in the tea, but transferred to the bath.

1. How does stirring affect the temperature of a cup of tea?

Stirring a cup of tea helps distribute the heat evenly throughout the liquid, causing the overall temperature to increase. This is due to convection, as the movement of the liquid brings the hotter portions to the surface while cooler portions sink to the bottom.

2. Why does the direction of stirring matter when making tea?

The direction of stirring can affect the taste and strength of the tea. Stirring in a circular motion creates a vortex, which helps to extract more flavor from the tea leaves. Stirring in a back and forth motion can create a more gentle extraction, resulting in a weaker tea.

3. Does stirring a cup of tea affect the rate of cooling?

Yes, stirring a cup of tea can actually help it cool down faster. As the liquid is stirred, it comes into contact with the cool air more frequently, allowing for more heat to be transferred from the tea to the surrounding environment.

4. Can stirring a cup of tea change its color?

Yes, stirring a cup of tea can cause it to change color. This is because stirring can release more tannins from the tea leaves, which can darken the color of the tea. Additionally, if milk or cream is added to the tea, stirring can help distribute it evenly and create a more uniform color.

5. How long should I stir a cup of tea for optimal flavor?

The optimal stirring time for a cup of tea can vary depending on personal preference. However, most experts recommend stirring for about 10-15 seconds. This allows for proper extraction of flavor and prevents over-stirring, which can cause the tea to become bitter.

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