One mole of a monatomic ideal gas is taken along the reversible path $P V^{2} = constant$ from $300 K$ to $360 K$ ( $R = 8.314 J mo l^{- 1} K^{- 1}$ ). For a polytropic path $P V^{n} = const$ , $C = C_{V} + R / (1 - n)$ . Which statements are correct?

AThe molar heat capacity along this path is $\frac{1}{2} R$ , about $4.16 J mo l^{- 1} K^{- 1}$ .
BThe heat absorbed by the gas is about $249 J$ .
C $Δ U$ of the gas is about $748 J$ .
DThe gas does about $499 J$ of work on the surroundings.

Answer: A, B, C

C = \frac{3}{2} R + R / (1 - 2) = \frac{1}{2} R = 4.157

(A true).

q = C Δ T = 249.4 J

(B true).

Δ U = \frac{3}{2} R (60) = 748.3 J

(C true). By the first law

w_{o n} = Δ U - q = 498.8 J

is done on the gas; since

V \propto 1/ T

along this path the gas is compressed on heating, so it does not do work on the surroundings (D false).