Thermodynamics Question (entropy and heat capacities)

[Jake4]

Homework Statement

Two somewhat related problems:

1. Using the expression for change in entropy of an ideal gas per mole:

ΔS=C_V+R ln(V)

Calculate the change in entropy, change in Helmholtz Free Energy and change in Gibbs Free Energy when 1 mole of an ideal gas is compressed from 1 atm to 20 atm at 20°Cand

2. An ideal gas has a molar specific heat given by C_V=A+BT where A and B are constants.
Using a thermodynamic identity find an expression for the change in entropy as it goes from V₁, T₁ to V₂ , T₂.

Homework Equations

dS= C_VdT/T

The Attempt at a Solution

My main issues with these problems (I may be totally missing something) is that C_V is given in both, yet they imply a change in Volume. By definition, C_V is the heat capacity at constant Volume. Is this something I should simply overlook to carry out the calculations?

Thank you so much for the help :)

 

[Jake4]

For the second problem, if I just go through the calculations I can integrate and find an expression for S, without even including V. Is that permissible even if in the problem it talks about going from V₁ to V₂ ?

 

[psmt]

1. Using the expression for change in entropy of an ideal gas per mole:

ΔS=C_V+R ln(V)

The expression for ΔS here can't be right: it should be a function of initial and final temperature, and initial and final volume (for fixed particle number). You can derive said function for the ideal gas by considering S=S(T,V), taking the differential of this, applying a Maxwell relation, and integrating. The result is on the Wikipedia 'Ideal gas' page. Fair enough, in the question the temperature stays fixed, but ΔS should still be a function of the initial and final volume of the gas.

For the second part, the general formula involving the initial and final temperatures could come in handy. Are you sure this gas is ideal, having a heat capacity that depends on temperature?

As a more general point, there's no reason why the heat capacity at constant volume can't appear in the solution to problems involving gases that change volume, as these questions illustrate.

 

[Andrew Mason]

The expression for ΔS here can't be right: it should be a function of initial and final temperature, and initial and final volume (for fixed particle number). You can derive said function for the ideal gas by considering S=S(T,V), taking the differential of this, applying a Maxwell relation, and integrating. The result is on the Wikipedia 'Ideal gas' page. Fair enough, in the question the temperature stays fixed, but ΔS should still be a function of the initial and final volume of the gas.

Where does it say that this is an isothermal process?

If it is isothermal (dT =0), dQr = PdV, so:

$$\Delta S = \int dS = \int dQ_r/T = \int (PdV)/T = \int RdV/V = R\ln(V_f/V_i)$$

For the second part, the general formula involving the initial and final temperatures could come in handy. Are you sure this gas is ideal, having a heat capacity that depends on temperature?

Heat capacities of non-monatomic ideal gases will change with temperature due to the fact that vibrational and rotational modes are not available at lower temperatures.

AM

 

[psmt]

Where does it say that this is an isothermal process?

If it is isothermal (dT =0), dQr = PdV, so:

$$\Delta S = \int dS = \int dQ_r/T = \int (PdV)/T = \int RdV/V = R\ln(V_f/V_i)$$

Heat capacities of non-monatomic ideal gases will change with temperature due to the fact that vibrational and rotational modes are not available at lower temperatures.

AM

The first part says "at 20 degrees C" - I'm taking that to mean the whole process takes place at that temp?

As for the variation of heat capacity with temperature, I'm guessing you're referring to the freezing out of certain modes and the transition to quantum behaviour? I forgot to say classical ideal gas, so good point.

 

[psmt]

Or, i can believe that if anharmonicity of bonds is taken into account for vibrational modes, then the heat capacity of the gas will vary with T even if the molecules are non-interacting, which i agree makes a classical ideal gas with a T-dependent heat capacity. Is this what you had in mind? (Damn these technicalities!)

 

[Andrew Mason]

Or, i can believe that if anharmonicity of bonds is taken into account for vibrational modes, then the heat capacity of the gas will vary with T even if the molecules are non-interacting, which i agree makes a classical ideal gas with a T-dependent heat capacity. Is this what you had in mind? (Damn these technicalities!)

I am not sure what you mean by "anharmonicity of bonds".

There is no such thing as an ideal diatomic gas that obeys only classical laws of physics. Temperature dependence of heat capacity of a diatomic ideal gas (ie. the gas obeys the ideal gas law: PV = nRT with molecules consisting of two covalently bonded atoms) requires quantum mechanics to explain.

AM

 

[psmt]

I am not sure what you mean by "anharmonicity of bonds".

I'm talking about the equipartition theorem: that each quadratic degree of freedom in the classical Hamiltonian contributes k/2 to the heat capacity, which is independent of temperature. By harmonic bond i mean a model in which the effective potential between two atoms in each gas molecule is truncated at quadratic order about the minimum, giving a constant contribution to the heat capacity. By anharmonic i mean including higher terms in the model, which presumably would lead to a T-dependent heat capacity within a purely classical context. Not that this is necessarily relevant or important, especially now we're clear that we're talking about QM.

There is no such thing as an ideal diatomic gas that obeys only classical laws of physics. Temperature dependence of heat capacity of a diatomic ideal gas (ie. the gas obeys the ideal gas law: PV = nRT with molecules consisting of two covalently bonded atoms) requires quantum mechanics to explain.

I agree. Of course, the internal structure of atoms of a monatomic gas will technically make a tiny contribution to the heat capacity too, if we're talking about real quasi-ideal gases... not sure if that would ever be measurable though.