Understanding the Reversible Adiabatic Process

A reversible adiabatic process is a thermodynamic process in which no heat is transferred into or out of the system (\(\Delta Q = 0\)), and the process is reversible, meaning it can be reversed without any change in the surroundings. For an ideal gas undergoing a reversible adiabatic process, the relationship between pressure (\(P\)) and volume (\(V\)) is given by:

\[PV^\gamma = \text{constant}\]

where \(\gamma\) is the adiabatic index, defined as the ratio of the specific heat at constant pressure (\(C_p\)) to the specific heat at constant volume (\(C_v\)), i.e., \(\gamma = C_p / C_v\).

Deriving the Temperature Ratio (\(\frac{{{T_1}}}{{{T_2}}}}\)) in Adiabatic Process

We need to find the ratio of initial temperature (\(T_1\)) to final temperature (\(T_2\)). We can use the ideal gas law, which states:

\[PV = nRT\]

where \(n\) is the number of moles and \(R\) is the ideal gas constant. From this equation, we can express pressure \(P\) as \(P = \frac{nRT}{V}\) or volume \(V\) as \(V = \frac{nRT}{P}\).

Using the P-V and Ideal Gas Relationship

Let's substitute the expression for \(P\) from the ideal gas law into the adiabatic process equation \(PV^\gamma = \text{constant}\):

\[\left(\frac{nRT}{V}\right)V^\gamma = \text{constant}\]

\[nRT V^{\gamma-1} = \text{constant}\]

Since \(nR\) is a constant, we get a relationship between temperature and volume:

\[TV^{\gamma-1} = \text{constant}\]

For two states (initial state 1 and final state 2) in the reversible adiabatic process, we have:

\[T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}\]

Rearranging this to find the ratio \(\frac{{{T_1}}}{{{T_2}}}}\):

\[\frac{{{T_1}}}{{{T_2}}}} = \frac{{{V_2^{\gamma-1}}}}{{{V_1^{\gamma-1}}}}} = \left(\frac{{{V_2}}}{{{V_1}}}}\right)^{\gamma-1}\]

Using \(v\) for volume as in the options, this is \(\left(\frac{{{v_2}}}{{{v_1}}}}\right)^{\gamma-1}\). This expression is equivalent to \(\left(\frac{{{v_1}}}{{{v_2}}}}\right)^{-(\gamma-1)}\), which is not in the given options in a matching form.

Using the P-V and Ideal Gas Relationship to find T-P relation

Now, let's substitute the expression for \(V\) from the ideal gas law into the adiabatic process equation \(PV^\gamma = \text{constant}\):

\[P\left(\frac{nRT}{P}\right)^\gamma = \text{constant}\]

\[P \left(\frac{(nR)^\gamma T^\gamma}{P^\gamma}\right) = \text{constant}\]

\[\frac{(nR)^\gamma T^\gamma}{P^{\gamma-1}} = \text{constant}\]

Since \((nR)^\gamma\) is a constant, we get a relationship between temperature and pressure:

\[\frac{{{T^\gamma}}}{{{P^{\gamma-1}}}}} = \text{constant}\]

For two states in the reversible adiabatic process:

\[\frac{{{T_1^\gamma}}}{{{P_1^{\gamma-1}}}}} = \frac{{{T_2^\gamma}}}{{{P_2^{\gamma-1}}}}}\]

Rearranging this to find the ratio \(\frac{{{T_1}}}{{{T_2}}}}\):

\[\frac{{{T_1^\gamma}}}{{{T_2^\gamma}}}} = \frac{{{P_1^{\gamma-1}}}}{{{P_2^{\gamma-1}}}}}\]

\[\left(\frac{{{T_1}}}{{{T_2}}}}\right)^\gamma = \left(\frac{{{P_1}}}{{{P_2}}}}\right)^{\gamma-1}\]

Taking the \(\gamma\)-th root of both sides:

\[\frac{{{T_1}}}{{{T_2}}}} = \left[\left(\frac{{{P_1}}}{{{P_2}}}}\right)^{\gamma-1}\right]^{1/\gamma} = \left(\frac{{{P_1}}}{{{P_2}}}}\right)^{\frac{{\gamma - 1}}{\gamma }}\]

Using \(p\) for pressure as in the options, this is \(\left(\frac{{{p_1}}}{{{p_2}}}}\right)^{\frac{{\gamma - 1}}{\gamma }}\).

Comparing Derived Formula with Options

Let's compare our derived formulas with the given options:

The derived formula for the temperature ratio in terms of pressure is \(\frac{{{T_1}}}{{{T_2}}}} = \left(\frac{{{p_1}}}{{{p_2}}}}\right)^{\frac{{\gamma - 1}}{\gamma }}\).

• Option 1: \(\left( {{v_1}{v_2}} \right)^{\frac{{\gamma - 1}}{\gamma }}\) - Does not match.
• Option 2: \(\left( {\frac{{{v_1}}}{{{v_2}}}} \right)^{\frac{{\gamma - 1}}{\gamma }}\) - Does not match our T-V relation \(\left(\frac{{{v_2}}}{{{v_1}}}}\right)^{\gamma-1}\).
• Option 3: \(\left( {\frac{{{p_1}}}{{{p_2}}}} \right)^{\frac{{\gamma - 1}}{\gamma }}\) - Matches our derived T-P relation.
• Option 4: \(\left( {\frac{{{p_2}}}{{{p_1}}}} \right)^\gamma \) - Does not match.

Therefore, the correct expression for the ratio \(\left( {\frac{{{T_1}}}{{{T_2}}}} \right)\) in a reversible adiabatic process is \(\left( {\frac{{{p_1}}}{{{p_2}}}} \right)^{\frac{{\gamma - 1}}{\gamma }}\).