Change of temperature in the decomposition of an ideal diatomic gas into a monoatomic one at constant volume

Question

Let's suppose we have $N$ molecules of a diatomic ideal gas at temperature $T_1$ and pressure $P_1$ inside a constant volume $V$. The system is isolated from the surroundings so that the internal energy remains constant. The diatomic gas reacts decomposing into a monoatomic one, according to this reaction: $A_2\rightarrow2A+Q$, where $Q$ is the heat released in the reaction (which is exothermic). I need to find out the expression of the final temperature of the system, $T_2$, once all the molecules of the diatomic gas have reacted.

I know that for ideal gases $U=nc_VT$, with $U$ the internal energy of the gas and $c_V$ its molar heat capacity, and that $c_V$ is different for diatomic and monoatomic gases. However, by combining these formulas I don't get the right solution.

How could this be solved?

Answer 1

The heat of reaction of an ideal gas, $\Delta H_R(T_0)$, is defined as the change in enthalpy in going from pure reactants to pure products, holding the temperature $T_0$ constant by adding an amount of heat Q equal to $Q=\Delta H_R(T_0)$. So, for an exothermic reaction, $\Delta H_R(T_0)$ is negative. For the same change from reactants to products at constant $T_0$, the change in internal energy is $$\Delta U_R(T_0)=\Delta H_R(T_0)-\Delta (PV)=\Delta H_R(T_0)-(\Delta n)RT_0$$In the present case, $\Delta n=1\ mole$.

For the case of a reaction carried out adiabatically at constant volume, we need to apply Hess' law to get the change in temperature. This leads to $$\Delta U=\Delta U_R(T_0)+n_pC_{vp}(T-T_0)=0$$ where $n_P$ is the number of moles of product in the reaction formula and $C_{vp}$ is the weighted average molar heat at capacity at constant volume of the product mixture. So, in the present case, $n_p=2$ and $C_{vp}$ is the molar heat capacity at constant volume of A. So, combining the above two equations, we would have: $$2C_{vA}(T-T_0)=-\Delta H_R(T_0)+RT_0$$ This development depends on the fact that, for an ideal gas, both U and H are independent of pressure and specific volume.