Thermodynamic Cycle 2: Understanding PV^y, Cp/Cv and PV=nRT

[subzero0137]

2. PV^y = const, y=Cp/Cv, PV=nRT, 3.

I've drawn the cycle for part a) of the question, but I'm not sure how to do part b). I know I need to relate Vm and 4Vo using the PV^y = constant relation but I'm stuck as to how to do it.

 

[haruspex]

I know I need to relate Vm and 4Vo using the PV^y = constant relation

Using that, write an equation relating P₀, V₀, P_m and V_m.

 

[subzero0137]

Using that, write an equation relating P₀, V₀, P_m and V_m.

Thanks, I think I got it now. Just to confirm, would P_m = P_0/32?

 

[haruspex]

Thanks, I think I got it now. Just to confirm, would P_m = P_0/32?

Yes.

 

[subzero0137]

Yes.

Sorry to bother you again but the next part of the questions asks to determine the work done, heat transfer and change in internal energy in each process. So if we consider the first, isobaric process then the work done on the gas is simply W_on = -P₀ * (4V₀ - V₀) = -P₀ * 3V₀. To calculate the heat transferred (Q) and change in internal energy (ΔU) I can use the first law of thermodynamics, but the equations for Q and ΔU involve variables that I don't know. ΔU=nC_VΔT and Q=nC_PΔT. I don't know how to get either Q or ΔU if I don't know n (moles) and change in temperature.

 

[Chestermiller]

Sorry to bother you again but the next part of the questions asks to determine the work done, heat transfer and change in internal energy in each process. So if we consider the first, isobaric process then the work done on the gas is simply W_on = -P₀ * (4V₀ - V₀) = -P₀ * 3V₀. To calculate the heat transferred (Q) and change in internal energy (ΔU) I can use the first law of thermodynamics, but the equations for Q and ΔU involve variables that I don't know. ΔU=nC_VΔT and Q=nC_PΔT. I don't know how to get either Q or ΔU if I don't know n (moles) and change in temperature.

Try it algebraically, also making use of the ideal gas law, and see how it plays out. I think you will be pleasantly surprised.

 

[subzero0137]

Do it "per mole."

ΔU=C_VΔT and Q=C_PΔT? But the temperature differences are still missing. I tried to do P₀V₀/T₁ = P₀4V₀/T₂ ⇒ T₂=4T₁ but that doesn't give a change in temperature

 

[Chestermiller]

$$nT_0=\frac{P_0V_0}{R}$$
$$nT_1=\frac{P_0(4V_0)}{R}$$
$$nC_v(T_1-T_0)=C_v\frac{3P_0V_0}{R}$$
For a monoatomic gas, what is the molar Cv in terms of R?

 

[subzero0137]

$$nT_0=\frac{P_0V_0}{R}$$
$$nT_1=\frac{P_0(4V_0)}{R}$$
$$nC_v(T_1-T_0)=C_v\frac{3P_0V_0}{R}$$
For a monoatomic gas, what is the molar Cv in terms of R?

Molar Cv = 3/2 R? Sorry I'm still confused

 

[Chestermiller]

Molar Cv = 3/2 R? Sorry I'm still confused

$$\Delta U=nC_v\Delta T=\left(\frac{3R}{2}\right)\frac{3P_0V_0}{R}=?$$