When the gas is ideal and process is isothermal, then

Options

A. `P_(1)V_(1) = P_(2)V_(2)`
B. `Delta U = 0`
C. `Delta W = 0`
D. `Delta H_(1) = Delta H_(2)`

Answer

To solve the question regarding the behavior of an ideal gas during an isothermal process, we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature

### Step 2: Define Isothermal Process
An isothermal process is one where the temperature remains constant. Therefore, the change in temperature (\( \Delta T \)) is zero:
\[ \Delta T = 0 \]

### Step 3: Analyze the Implications of Isothermal Conditions
Since \( T \) is constant and \( n \) (number of moles) is constant, the product \( PV \) must also remain constant. Thus, we can express this as:
\[ P_1 V_1 = P_2 V_2 \]
This confirms that the first option is correct.

### Step 4: Calculate Change in Internal Energy (\( \Delta U \))
The change in internal energy for an ideal gas is given by:
\[ \Delta U = nC_V \Delta T \]
Since \( \Delta T = 0 \):
\[ \Delta U = nC_V \times 0 = 0 \]
Thus, the second option is also correct.

### Step 5: Analyze Work Done (\( \Delta W \))
The work done in a thermodynamic process can be related to the change in internal energy and enthalpy. According to the first law of thermodynamics:
\[ \Delta U = \Delta H - \Delta W \]
Given that \( \Delta U = 0 \):
\[ 0 = \Delta H - \Delta W \]
This implies:
\[ \Delta W = \Delta H \]
Thus, the work done is not zero, making the third option incorrect.

### Step 6: Analyze Change in Enthalpy (\( \Delta H \))
The change in enthalpy is given by:
\[ \Delta H = nC_P \Delta T \]
Since \( \Delta T = 0 \):
\[ \Delta H = nC_P \times 0 = 0 \]
This means:
\[ \Delta H_2 - \Delta H_1 = 0 \]
or:
\[ \Delta H_2 = \Delta H_1 \]
Thus, the fourth option is correct.

### Conclusion
The correct options are:
- Option 1: Correct
- Option 2: Correct
- Option 3: Incorrect
- Option 4: Correct

### Summary of Correct Options
The correct options are 1, 2, and 4.

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