When an ideal diatomic gas is heated at constant pressure fraction of the heat energy supplied which increases the internal energy of the gas is

Options

A. `(2)/(5)`
B. `(3)/(5)`
C. `(3)/(7)`
D. `(5)/(7)`

Explanation

To solve the problem of determining the fraction of heat energy supplied that increases the internal energy of an ideal diatomic gas heated at constant pressure, we can follow these steps:

### Step 1: Understand the Given Information
We are dealing with an ideal diatomic gas that is heated at constant pressure. We need to find the fraction of the heat energy supplied that contributes to the increase in the internal energy of the gas.

### Step 2: Use the Formula for Heat Supplied
At constant pressure, the heat supplied \( Q \) can be expressed as:
\[ Q = n C_p \Delta T \]
where:
- \( n \) = number of moles of gas
- \( C_p \) = molar specific heat at constant pressure
- \( \Delta T \) = change in temperature

### Step 3: Use the Formula for Change in Internal Energy
The change in internal energy \( \Delta U \) for an ideal gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where:
- \( C_v \) = molar specific heat at constant volume

### Step 4: Identify the Values of \( C_p \) and \( C_v \) for a Diatomic Gas
For a diatomic gas:
- \( C_p = \frac{7}{2} R \)
- \( C_v = \frac{5}{2} R \)

### Step 5: Substitute \( C_p \) and \( C_v \) into the Equations
Now, substituting these values into our equations for \( Q \) and \( \Delta U \):
\[ Q = n \left(\frac{7}{2} R\right) \Delta T \]
\[ \Delta U = n \left(\frac{5}{2} R\right) \Delta T \]

### Step 6: Calculate the Fraction of Heat Energy that Increases Internal Energy
To find the fraction of heat energy that goes into increasing the internal energy, we can take the ratio:
\[ \frac{\Delta U}{Q} = \frac{n \left(\frac{5}{2} R\right) \Delta T}{n \left(\frac{7}{2} R\right) \Delta T} \]
The \( n \), \( R \), and \( \Delta T \) cancel out:
\[ \frac{\Delta U}{Q} = \frac{\frac{5}{2}}{\frac{7}{2}} = \frac{5}{7} \]

### Step 7: Conclusion
Thus, the fraction of the heat energy supplied that increases the internal energy of the gas is:
\[ \frac{5}{7} \]

### Final Answer
The answer is \( \frac{5}{7} \).

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