If E is the energy density in an ideal gas, then the pressure of the ideal gas is

Options

A. `P = (2)/(3) E`
B. `P = (3)/(2) E`
C. `P = (5)/(2) E`
D. `P = (2)/(5) E`

Answer

To find the pressure of an ideal gas in terms of its energy density \( E \), we can follow these steps:

### Step 1: Understand Energy Density
Energy density \( E \) is defined as the energy per unit volume of the gas. For an ideal gas, the total kinetic energy can be expressed in terms of the number of molecules, temperature, and the volume.

### Step 2: Relate Pressure to Kinetic Energy
The pressure \( P \) exerted by an ideal gas can be related to the kinetic energy of the gas molecules. The formula for pressure in terms of the root mean square velocity \( V_{\text{rms}} \) and density \( \rho \) is given by:
\[ P = \frac{1}{3} \rho V_{\text{rms}}^2 \]

### Step 3: Express Density in Terms of Volume
The density \( \rho \) can be expressed as:
\[ \rho = \frac{m}{V} \]
where \( m \) is the mass of the gas and \( V \) is the volume.

### Step 4: Substitute Density into Pressure Equation
Substituting the expression for density into the pressure equation gives:
\[ P = \frac{1}{3} \left(\frac{m}{V}\right) V_{\text{rms}}^2 \]

### Step 5: Relate Kinetic Energy to Energy Density
The kinetic energy per molecule is given by:
\[ \text{K.E.} = \frac{1}{2} m V_{\text{rms}}^2 \]
The total kinetic energy in the volume \( V \) is:
\[ \text{Total K.E.} = \frac{1}{2} m V_{\text{rms}}^2 \]
The energy density \( E \) can be expressed as:
\[ E = \frac{\text{Total K.E.}}{V} = \frac{1}{2} \frac{m}{V} V_{\text{rms}}^2 = \frac{1}{2} \rho V_{\text{rms}}^2 \]

### Step 6: Substitute Energy Density into Pressure Equation
From the expression for energy density, we can rearrange it to find \( \rho V_{\text{rms}}^2 \):
\[ \rho V_{\text{rms}}^2 = 2E \]
Substituting this back into the pressure equation gives:
\[ P = \frac{1}{3} \left(\frac{2E}{\frac{1}{2}}\right) = \frac{2}{3} E \]

### Conclusion
Thus, the pressure \( P \) of the ideal gas in terms of energy density \( E \) is:
\[ P = \frac{2}{3} E \]

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