For `H_(2) gas, C_(p) - C_(v) = a` and for `O_(2) gas C_(p) - C_(v) = b`, so the relation between a and b is

Options

A. a = 16 b
B. 16a = b
C. a = 4b
D. a = b

Answer

To find the relation between \( a \) and \( b \) for the specific heat capacities of hydrogen and oxygen gases, we can follow these steps:

### Step 1: Understand the relationship between \( C_p \) and \( C_v \)
The difference between the specific heat capacities at constant pressure (\( C_p \)) and constant volume (\( C_v \)) for any ideal gas is given by the equation:
\[ C_p - C_v = R \]
where \( R \) is the universal gas constant.

### Step 2: Express \( a \) and \( b \) in terms of \( R \)
For hydrogen gas (\( H_2 \)):
\[ C_p - C_v = a = \frac{R}{M_{H_2}} \]
where \( M_{H_2} \) (molecular mass of hydrogen) = 2 g/mol. Thus,
\[ a = \frac{R}{2} \]

For oxygen gas (\( O_2 \)):
\[ C_p - C_v = b = \frac{R}{M_{O_2}} \]
where \( M_{O_2} \) (molecular mass of oxygen) = 32 g/mol. Thus,
\[ b = \frac{R}{32} \]

### Step 3: Find the relation between \( a \) and \( b \)
Now, we can express the relationship between \( a \) and \( b \):
\[ \frac{a}{b} = \frac{\frac{R}{2}}{\frac{R}{32}} = \frac{R}{2} \times \frac{32}{R} = \frac{32}{2} = 16 \]
This implies:
\[ a = 16b \]

### Conclusion
Thus, the relation between \( a \) and \( b \) is:
\[ a = 16b \]