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

Answer

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

### Step 1: Understand the relationship between \( C_p \), \( C_v \), and \( R \)

The relationship between the molar heat capacities at constant pressure \( C_p \) and constant volume \( C_v \) is given by:

\[ C_p - C_v = R \]

Where:
- \( C_p \) = Molar heat capacity at constant pressure
- \( C_v \) = Molar heat capacity at constant volume
- \( R \) = Gas constant

### Step 2: Substitute the given values

From the problem, 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:

\[ R = C_p - C_v \]

\[ R = 0.245 - 0.165 \]

### Step 3: Calculate \( R \)

Calculating the above expression:

\[ R = 0.080 \, \text{cal g}^{-1} \, \text{°C}^{-1} \]

### Step 4: Convert \( R \) to ergs

We need to convert \( R \) from calories to ergs using the conversion factor provided:

\[ 1 \, \text{cal} = 4.2 \times 10^7 \, \text{erg} \]

Thus,

\[ R = 0.080 \, \text{cal g}^{-1} \, \text{°C}^{-1} \times 4.2 \times 10^7 \, \text{erg cal}^{-1} \]

### Step 5: Perform the multiplication

Calculating the above expression:

\[ R = 0.080 \times 4.2 \times 10^7 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]

\[ R = 0.336 \times 10^7 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]

### Step 6: Finalize the result

We can express this in scientific notation:

\[ R = 3.36 \times 10^6 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]

### Conclusion

The gas constant \( R \) for 1 g of gas is:

\[ R = 3.36 \times 10^6 \, \text{erg g}^{-1} \, \text{°C}^{-1} \]

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