5 mole of an ideal gas at temp. T are compressed isothermally from 12 atm. To 24 atm. Calculate the value of 10 r. Where, `r=("Work done along reversible process")/("Work done along single step irreversible process")" (Given : ln 2 = 0.7)"`

Answer

To solve the problem, we need to calculate the value of \(10R\), where \(R\) is defined as the ratio of the work done along a reversible process to the work done along a single-step irreversible process. We will follow these steps:

### Step 1: Identify the Given Information
- Number of moles, \(n = 5\) moles
- Initial pressure, \(P_1 = 12\) atm
- Final pressure, \(P_2 = 24\) atm
- Temperature, \(T\) (constant during isothermal process)
- Given: \(\ln 2 = 0.7\)

### Step 2: Calculate Work Done in Reversible Process
The work done during an isothermal reversible process can be calculated using the formula:
\[ W_{\text{rev}} = -nRT \ln \left(\frac{P_1}{P_2}\right) \]
Substituting the values we have:
\[ W_{\text{rev}} = -5RT \ln \left(\frac{12}{24}\right) = -5RT \ln \left(\frac{1}{2}\right) = -5RT (-\ln 2) = 5RT \ln 2 \]
Using the given value of \(\ln 2 = 0.7\):
\[ W_{\text{rev}} = 5RT \times 0.7 = 3.5RT \]

### Step 3: Calculate Work Done in Irreversible Process
The work done during a single-step irreversible process can be calculated using the formula:
\[ W_{\text{irrev}} = -P_{\text{ext}} \Delta V \]
For isothermal processes, we can express \(\Delta V\) in terms of pressures:
\[ \Delta V = \frac{nRT}{P_1} - \frac{nRT}{P_2} \]
Thus, we can write:
\[ W_{\text{irrev}} = -P_{\text{ext}} \left(\frac{nRT}{P_2} - \frac{nRT}{P_1}\right) \]
Here, we take \(P_{\text{ext}} = P_2 = 24\) atm:
\[ W_{\text{irrev}} = -24 \left(\frac{5RT}{24} - \frac{5RT}{12}\right) \]
Calculating the difference:
\[ \frac{5RT}{24} - \frac{5RT}{12} = \frac{5RT}{24} - \frac{10RT}{24} = -\frac{5RT}{24} \]
Thus:
\[ W_{\text{irrev}} = -24 \left(-\frac{5RT}{24}\right) = 5RT \]

### Step 4: Calculate \(R\)
Now we can find \(R\):
\[ R = \frac{W_{\text{rev}}}{W_{\text{irrev}}} = \frac{3.5RT}{5RT} = \frac{3.5}{5} = 0.7 \]

### Step 5: Calculate \(10R\)
Finally, we calculate \(10R\):
\[ 10R = 10 \times 0.7 = 7 \]

### Final Answer
Thus, the value of \(10R\) is:
\[ \boxed{7} \]