Two different answers for work done during compression

[etotheipi]

Homework Statement

Calculating work done during adiabatic compression (please see problem statement below)

Relevant Equations

Ideal gas laws, internal energy

I'm having a little trouble with part a) of this question:

Since it is stated that the heating is slow, I thought it was reasonable to assume the process is reversible which means that the pressure in both sides should be equal. Consequently, $$W = - \int_{V_{0}}^{V_{1}} P dV = - \int_{V_{0}}^{V_{1}} kV^{- \gamma} dV = \frac{1}{\gamma-1}(P_{1}V_{1} - P_{0}V_{0})$$ For $\gamma = 1.5$, the outside coefficient is $2$.

However, if I use the first law (with $Q=0$ due to the insulated/adiabatic condition), I get $$W = \Delta U = \frac{3}{2}(nRT_{1} - nRT_{0}) = \frac{3}{2}(P_{1}V_{1} - P_{0}V_{0})$$I can't figure out why the two coefficients of $(P_{1}V_{1} - P_{0}V_{0})$ are different!

After this is sorted out, I assume it will just be a case of doing some rearrangement (i.e. w/ $\frac{P_{0}V_{0}}{T_{0}} = \frac{P_{1}V_{1}}{T_{1}}$ etc.) to get it into the required form?

 

[TSny]

However, if I use the first law (with $Q=0$ due to the insulated/adiabatic condition), I get $$W = \Delta U = \frac{3}{2}(nRT_{1} - nRT_{0}) = \frac{3}{2}(P_{1}V_{1} - P_{0}V_{0})$$I can't figure out why the two coefficients of $(P_{1}V_{1} - P_{0}V_{0})$ are different!

The coefficient $\frac{3}{2}$ in $\Delta U$ is only for a monatomic ideal gas. Is the gas monatomic in this problem?

 

[etotheipi]

The coefficient $\frac{3}{2}$ in $\Delta U$ is only for a monatomic ideal gas. Is the gas monatomic in this problem?

Thank you, I just googled $\gamma$ for an ideal monatomic gas (it turns out to be roughly 1.66), which gives $\frac{1}{\gamma-1} \approx 1.5$, as was obtained via the other method!

 

[TSny]

Thank you, I just googled $\gamma$ for an ideal monatomic gas (it turns out to be roughly 1.66), which gives $\frac{1}{\gamma-1} \approx 1.5$, as was obtained via the other method!

Is everything OK now?

For a general ideal gas, $U = nC_VT$ and $C_V = \frac{R}{\gamma - 1}$.