Two containers of equal volume contain the same gas at pressures P₁ and P₂ and absolute temperatures T₁ and T₂ respectively. On joining the vessels, the gas reaches a common pressure P and common temperature T. The ratio P/T is equal to

To find the ratio $\frac{P}{T}$ after the two vessels are connected, we can use the ideal gas law and the principle of conservation of mass (or total number of moles of the gas).

Let the volume of each of the two containers be $V$.
Initially, the two containers are separate:
For Container 1: Pressure is $P_1$, Temperature is $T_1$, and Volume is $V$.
Using the ideal gas equation $PV=nRT$, the number of moles $n_1$ in the first container is:
$n_1=\frac{P_{1}V}{R{T}_1}$

For Container 2: Pressure is $P_2$, Temperature is $T_2$, and Volume is $V$.
The number of moles $n_2$ in the second container is:
$n_2=\frac{P_{2}V}{R{T}_2}$

The total number of moles of gas in the combined system, $n_{\text{total}}$, is the sum of the moles in both containers:
$n_{\text{total}}=n_1+n_2=\frac{P_{1}V}{R{T}_1}+\frac{P_{2}V}{R{T}_2}$

When the two vessels are joined, the total volume of the system becomes:
$V_{\text{total}}=V+V=2V$

Let the common pressure be $P$ and the common temperature be $T$ when equilibrium is reached. The total number of moles can also be written in terms of these final parameters:
$n_{\text{total}}=\frac{P\left(2V\right)}{RT}$

Equating the two expressions for $n_{\text{total}}$:
$\frac{2PV}{RT}=\frac{P_{1}V}{R{T}_1}+\frac{P_{2}V}{R{T}_2}$

We can divide both sides of the equation by $\frac{V}{R}$:
$\frac{2P}{T}=\frac{P_1}{T_1}+\frac{P_2}{T_2}$

Dividing by 2 to solve for the ratio $\frac{P}{T}$ gives:
$\frac{P}{T}=\frac{P_1}{2T_1}+\frac{P_2}{2T_2}$

Thus, the correct option is P₁/2T₁ + P₂/2T₂.