# Calculating final pressure in irreversible adiabatic compression

I am trying to solve the following question.

A gas is enclosed in a cylindrical can fitted with a piston. The walls of the can and the piston are adiabatic. The initial pressure, volume and temperature of the gas are 100 kPa, 400 $$cm^3$$ and 300 K respectively.The ratio of the specific heat capacities of the gas is C$$_p$$ / C$$_v$$ = 1.5. Find the pressure and temperature of the gas if it is suddenly compressed to 100 $$cm^3$$.

I tried to solve it by equating dU and δW. Since the gas is suddenly compressed it is an irreversible adiabatic change. δW = −$$p_2dV$$ and dU = $$nC_vdT$$. $$p_2$$ is the final pressure, $$n$$ is the number of moles present and $$C_v$$ is the molar specific heat capacity. (I followed the method given in this answer.)

$$nC_v(T_2−T_1) = −p_2(V_2−V_1)$$ From ideal gas equation $$p_2 = \frac{nRT_2}{V_2}$$. From the ratio given in the question, $$C_v$$ comes out to be $$2R$$. $$2nR(T_2−T_1) = −\frac{nRT_2(V_2−V_1)}{V_2}$$ After dividing both sides by $$nR$$ and substituting values of $$V_1$$ and $$V_2$$, the equation becomes
$$2T_2−2T_1=3T_2$$ This equation is wrong. I have copied everything except the value of $$p_2$$ from the answer linked above so I think I made some mistake in it. Please tell why my method is wrong and how will the question be solved ?

If you solve your equation algebraically, you get $$T_2=\frac{2T_1}{\left[3-\frac{V_1}{V_2}\right]}$$At $$V_1/V_2=2$$, $$T_2=2T_1$$ and at $$V_1/V_2=2.5$$, $$T_2=4T_1$$, but at $$V_1/V_2=3$$, $$T_2$$ becomes infinite. So it is not possible to compress the gas to a greater compression ratio than 3 without the pressure and temperature becoming infinite (at least not by applying a constant external pressure). In your problem statement, the compression ratio is 4.