Calculate the gas constant for 1 g of gas from the following data :
`C_p=0.245 "cal g"^(-1).^@C^(-1), C_v=0.165" cal g"^(-1).^@C^(-1)and J=4.2 xx10^7 erg cal^(-1)`

Options

A. `3.36xx10^6"erg g"^(-1 ).^@C^(-1)`
B. `2.13xx10^6"erg g"^(-1 ).^@C^(-1)`
C. `4.26xx10^6"erg g"^(-1 ).^@C^(-1)`
D. `4.57xx10^6"erg g"^(-1 ).^@C^(-1)`

Answer

To calculate the gas constant \( R \) for 1 g of gas using the given data, we can follow these steps:

### Step 1: Understand the relationship between \( C_p \), \( C_v \), and \( R \)
The relationship between the molar specific heats at constant pressure \( C_p \), constant volume \( C_v \), and the gas constant \( R \) is given by:
\[ R = C_p - C_v \]

### Step 2: Substitute the values of \( C_p \) and \( C_v \)
From the question, we have:
- \( C_p = 0.245 \, \text{cal g}^{-1} \, \text{°C}^{-1} \)
- \( C_v = 0.165 \, \text{cal g}^{-1} \, \text{°C}^{-1} \)

Now, substituting these values into the equation:
\[ R = 0.245 - 0.165 \]

### Step 3: Calculate \( R \)
Perform the subtraction:
\[ R = 0.245 - 0.165 = 0.080 \, \text{cal g}^{-1} \, \text{°C}^{-1} \]

### Step 4: Convert \( R \) from calories to ergs
We need to convert the value of \( R \) from calories to ergs using the conversion factor:
\[ 1 \, \text{cal} = 4.2 \times 10^7 \, \text{erg} \]

Thus, we multiply \( R \) by \( 4.2 \times 10^7 \):
\[ R = 0.080 \, \text{cal g}^{-1} \, \text{°C}^{-1} \times 4.2 \times 10^7 \, \text{erg cal}^{-1} \]

### Step 5: Calculate the final value of \( R \)
Calculating the multiplication:
\[ R = 0.080 \times 4.2 \times 10^7 = 0.336 \times 10^7 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]

### Final Answer
Thus, the gas constant \( R \) for 1 g of gas is:
\[ R = 3.36 \times 10^6 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]