One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_p, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_p, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_A/T_B$?

Step 1: Analyze the isothermal process.

For an isothermal expansion, the temperature remains constant throughout the process.

$$T_B = T_p$$

Step 2: Apply the adiabatic relation.

For an adiabatic expansion, the relation between temperature and volume is constant.

$$T V^{\gamma-1} = \text{constant}$$

Step 3: Relate the initial and final states of the adiabatic process.

Apply the relation to the initial state $(T_A, V_0)$ and the final state $(T_p, 5V_0)$.

$$T_A V_0^{\gamma-1} = T_p (5V_0)^{\gamma-1}$$

Step 4: Solve for the initial temperature $T_A$.

Simplify the equation by dividing both sides by $V_0^{\gamma-1}$.

$$T_A = T_p \cdot 5^{\gamma-1}$$

Step 5: Calculate the ratio of the initial temperatures.

Substitute the expressions for $T_A$ and $T_B$ to find the ratio.

$$\frac{T_A}{T_B} = \frac{T_p \cdot 5^{\gamma-1}}{T_p}$$ $$\frac{T_A}{T_B} = 5^{\gamma-1}$$

Answer: $\dfrac{T_A}{T_B} = 5^{\gamma-1}$