The pressure of an ideal gas is written as `p=(2E)/(3V)`.Here E refers to

Options

A. (a)translational kinetic energy
B. (b)rotational kinetic energy
C. (c)vibrational kinetic energy
D. (d)total kinetic energy.

Answer

To solve the question "The pressure of an ideal gas is written as \( p = \frac{2E}{3V} \). Here, E refers to what?" we will follow these steps:

### Step 1: Understand the given equation
We are given the equation for pressure \( p \) in terms of energy \( E \) and volume \( V \):
\[ p = \frac{2E}{3V} \]

### Step 2: Rearrange the equation
We can rearrange this equation to express energy \( E \) in terms of pressure \( p \) and volume \( V \):
\[ pV = \frac{2E}{3} \]
Multiplying both sides by \( \frac{3}{2} \) gives:
\[ E = \frac{3}{2} pV \]

### Step 3: Relate pressure to the ideal gas law
From the ideal gas law, we know that:
\[ pV = nRT \]
where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

### Step 4: Substitute the ideal gas law into the energy equation
Substituting \( pV \) from the ideal gas law into our expression for \( E \):
\[ E = \frac{3}{2} (nRT) \]

### Step 5: Relate energy to degrees of freedom
According to the equipartition theorem, the total energy \( E \) of an ideal gas can also be expressed as:
\[ E = \frac{F}{2} k_B T \]
where \( F \) is the number of degrees of freedom, and \( k_B \) is the Boltzmann constant.

### Step 6: Identify the degrees of freedom for an ideal gas
For a monatomic ideal gas, the number of degrees of freedom \( F \) is 3 (corresponding to motion in the x, y, and z directions). Thus, we can express the energy as:
\[ E = \frac{3}{2} k_B T \]

### Step 7: Conclusion
From our calculations, we see that the energy \( E \) refers to the translational kinetic energy of the gas particles. Therefore, the correct answer is that \( E \) refers to the translational kinetic energy.

### Final Answer
E refers to the translational kinetic energy.

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