Enthalpy of 1 mole monoatomic ideal gas is equals to :-

Options

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

Explanation

To find the enthalpy of 1 mole of a monoatomic ideal gas, we can follow these steps:

### Step 1: Understand the definition of enthalpy
Enthalpy (H) is defined as:
\[ H = U + PV \]
where \(U\) is the internal energy, \(P\) is the pressure, and \(V\) is the volume.

### Step 2: Express internal energy for an ideal gas
For an ideal gas, the internal energy \(U\) can be expressed as:
\[ U = n C_v \Delta T \]
where \(n\) is the number of moles, \(C_v\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature.

### Step 3: Express the PV term
The term \(PV\) can be expressed using the ideal gas law:
\[ PV = nRT \]
where \(R\) is the universal gas constant and \(T\) is the temperature.

### Step 4: Combine the expressions for enthalpy
Substituting the expressions for \(U\) and \(PV\) into the enthalpy equation gives:
\[ H = n C_v \Delta T + n R \Delta T \]
Factoring out \(n \Delta T\) gives:
\[ H = n (C_v + R) \Delta T \]

### Step 5: Relate \(C_p\) and \(C_v\)
For an ideal gas, the relationship between \(C_p\) (molar heat capacity at constant pressure) and \(C_v\) is given by:
\[ C_p = C_v + R \]
Thus, we can rewrite the enthalpy equation as:
\[ H = n C_p \Delta T \]

### Step 6: Substitute values for a monoatomic ideal gas
For a monoatomic ideal gas, it is known that:
\[ \frac{C_p}{C_v} = \frac{5}{3} \]
From this, we can express \(C_v\) in terms of \(C_p\):
\[ C_v = \frac{3}{5} C_p \]

### Step 7: Substitute \(C_v\) into the equation
Using the relation \(C_p - C_v = R\):
\[ C_p - \frac{3}{5} C_p = R \]
This simplifies to:
\[ \frac{2}{5} C_p = R \quad \Rightarrow \quad C_p = \frac{5}{2} R \]

### Step 8: Substitute \(C_p\) back into the enthalpy equation
Now substituting \(C_p\) back into the enthalpy equation:
\[ H = n \left(\frac{5}{2} R\right) \Delta T \]
For 1 mole of gas (\(n = 1\)):
\[ H = \frac{5}{2} R \Delta T \]

### Step 9: Final expression for enthalpy
Thus, the enthalpy of 1 mole of a monoatomic ideal gas is:
\[ H = \frac{5}{2} R T \]

### Final Answer
The enthalpy of 1 mole of a monoatomic ideal gas is:
\[ \frac{5}{2} R T \]