Heat and Thermodynamics

We know that in adiabatic expansion, there is no heat exchange. So, $\Delta Q = 0$.From the first law of thermodynamics:$\Delta Q = \Delta U + W$ Since $\Delta Q = 0$, this means: $0 = \Delta U + W$ So, $W = -\Delta U$The change in internal energy for $n$ moles of a diatomic gas is: $\Delta U = n C_V \Delta T$ where $C_V$ for a diatomic gas is $\frac{5}{2} R$.Here, $n = 3$ moles and the temperature change $\Delta T = -50^{\circ} \mathrm{C}$ (or -50 in Kelvin, since the size of the degree is the same).So, $W = -\Delta U = -n C_V \Delta T = -3 \times \frac{5}{2} R \times (-50)$Multiplying out: $-3 \times \frac{5}{2} R \times (-50) = 7.5 R \times 50 = 375 R$The work done by the gas is $375 R$.