Enthelpy & Internal Energy Change relation with Cp & Cv

[cooldudeachyut]

Homework Statement

Match the following
Given : Processes do not include chemical reactions. Assume C_P,m and C_V,m are independent of temperature for given substance and consider only pressure-volume work in given all processes.

Homework Equations

ΔU = Q - W
ΔH = ΔU + Δ(PV)
PV = nRT

The Attempt at a Solution

The answer's given as following -
A : P Q S
B : P Q R
C : P Q R S T
D : P R T

I can get all relations correct except I don't understand why A : R and B : S aren't right. Since it says "any substance" I think those options should be right even if they only work for an ideal gas.

For an isochoric process ΔV = 0, so
ΔU = Q = nC_V,mΔT
ΔH = ΔU + VΔP
VΔP = nRΔT

which gives the relation as,
ΔH = n(C_V,m + R)ΔT

Since C_P,m - C_V,m = R, I can conclude that
ΔH = nC_P,mΔT

Similarly for an isobaric process ΔP = 0, so
ΔU = Q - PΔV , where Q = nC_P,mΔT
PΔV = nRΔT

which gives the relation as,
ΔU = n(C_P,m - R)ΔT

and just like before it becomes,
ΔU = nC_V,mΔT

I also remember reading somewhere that equations in option A and B work for any process, so maybe relations A : T and B : T are correct as well but I cannot derive for them.

 

[dRic2]

I Think the exercise in unclear. If cp and cv are independent of T, then ΔU = nCvΔT and ΔH = nCpΔT are always correct. This comes from the very definition of them both:

$c_v = \left( \frac {\partial U} {\partial T} \right)_V$
$c_p = \left( \frac {\partial H} {\partial T} \right)_P$

 

[Chestermiller]

I agree with dRic2. This is a very poorly worded question. It seems to me, A and B are correct for all cases, except a phase change (T). C is the definition of enthalpy change, so it is always correct. D is true for a constant pressure process (R) and a phase change at constant pressure (T).

 

[cooldudeachyut]

Thanks for the help everyone.

I Think the exercise in unclear. If cp and cv are independent of T, then ΔU = nCvΔT and ΔH = nCpΔT are always correct. This comes from the very definition of them both:

$c_v = \left( \frac {\partial U} {\partial T} \right)_V$
$c_p = \left( \frac {\partial H} {\partial T} \right)_P$

If these molar heat capacities being independent of temperature make equations from A & B true for all processes by definition why don't they apply to state changes, i.e., option T?

 

[dRic2]

Because, as you can see, heat capacity is defined as the variation of internal energy or enthalpy with respect to temperature change. In a phase transition $ΔT = 0$ so heat capacity has to go to infinity $c→\infty$. This means that, since we don't have a temperature change, our definition breaks down