The user wants to find the ratio of initial temperatures, $T_{\mathrm{A}} / T_{\mathrm{B}}$, for two different thermodynamic processes involving one mole of an ideal gas, both ending in the same final state.

Step 1: Analyze the Adiabatic Expansion

For the first process, one mole of an ideal gas expands adiabatically from an initial state $(T_{\mathrm{A}}, V_0)$ to a final state $(T_{\mathrm{f}}, 5 V_0)$.

The relationship between temperature ($T$) and volume ($V$) for a reversible adiabatic process is given by: $T V^{\gamma-1} = \text{constant}$ where $\gamma = C_{\mathrm{p}} / C_{\mathrm{v}}$ is the ratio of specific heats.

Applying this relation to the initial and final states of the adiabatic process:

• Initial state: $T_1 = T_{\mathrm{A}}$, $V_1 = V_0$
• Final state: $T_2 = T_{\mathrm{f}}$, $V_2 = 5 V_0$

We can write: $T_{\mathrm{A}} V_0^{\gamma-1} = T_{\mathrm{f}} (5 V_0)^{\gamma-1}$

Let's simplify this equation: $T_{\mathrm{A}} V_0^{\gamma-1} = T_{\mathrm{f}} \cdot 5^{\gamma-1} \cdot V_0^{\gamma-1}$

Canceling the $V_0^{\gamma-1}$ term from both sides, we get a relation between $T_{\mathrm{A}}$ and $T_{\mathrm{f}}$: $T_{\mathrm{A}} = T_{\mathrm{f}} \cdot 5^{\gamma-1} \quad \cdots (1)$

Step 2: Analyze the Isothermal Expansion

For the second process, one mole of the same gas expands isothermally from an initial state $(T_{\mathrm{B}}, V_0)$ to the same final state $(T_{\mathrm{f}}, 5 V_0)$.

An isothermal process is defined as a process that occurs at a constant temperature. Therefore, the initial temperature must be equal to the final temperature.

• Initial temperature: $T_{\mathrm{B}}$
• Final temperature: $T_{\mathrm{f}}$

So, for the isothermal process, we have: $T_{\mathrm{B}} = T_{\mathrm{f}} \quad \cdots (2)$

Step 3: Calculate the Ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$

Now we have two equations:

1. $T_{\mathrm{A}} = T_{\mathrm{f}} \cdot 5^{\gamma-1}$
2. $T_{\mathrm{B}} = T_{\mathrm{f}}$

To find the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$, we can substitute the value of $T_{\mathrm{f}}$ from equation (2) into equation (1): $T_{\mathrm{A}} = T_{\mathrm{B}} \cdot 5^{\gamma-1}$

Now, we can find the required ratio by dividing both sides by $T_{\mathrm{B}}$: $\frac{T_{\mathrm{A}}}{T_{\mathrm{B}}} = 5^{\gamma-1}$

Step 4: Compare with the Options

The calculated ratio is $5^{\gamma-1}$. Comparing this with the given options:

A: $5^{\gamma-1}$ B: $5^{1-\gamma}$ C: $5^\gamma$ D: $5^{1+\gamma}$

The result matches option A.