Which has maximum internal energy at 298K?

Options

A. helium gas
B. oxygen gas
C. ozone gas
D. equal

Answer

To determine which substance has the maximum internal energy at 298 K among helium gas, oxygen gas, and ozone (O3), we will follow these steps:

### Step 1: Understand the relationship between internal energy and heat capacity
The change in internal energy (ΔU) of a gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar heat capacity at constant volume
- \( \Delta T \) = change in temperature

Since we are comparing the internal energies at a constant temperature (298 K), we can focus on the heat capacities \( C_v \) of the gases.

### Step 2: Calculate \( C_v \) for each gas
1. **For Helium (He)**:
- Helium is a monoatomic gas. The formula for \( C_v \) for a monoatomic ideal gas is:
\[ C_v = \frac{3}{2} R \]
- Therefore, for helium:
\[ C_v = \frac{3}{2} R = 1.5 R \]

2. **For Oxygen (O2)**:
- Oxygen is a diatomic gas. The formula for \( C_v \) for a diatomic ideal gas is:
\[ C_v = \frac{5}{2} R \]
- Therefore, for oxygen:
\[ C_v = \frac{5}{2} R = 2.5 R \]

3. **For Ozone (O3)**:
- Ozone is a triatomic molecule and has a more complex structure. The formula for \( C_v \) for a triatomic gas (bent) is:
\[ C_v = \frac{6}{2} R = 3 R \]
- Therefore, for ozone:
\[ C_v = 3 R \]

### Step 3: Compare the \( C_v \) values
Now, we compare the \( C_v \) values calculated:
- For Helium: \( C_v = 1.5 R \)
- For Oxygen: \( C_v = 2.5 R \)
- For Ozone: \( C_v = 3 R \)

### Step 4: Determine which has the maximum internal energy
Since internal energy is directly proportional to \( C_v \), the gas with the highest \( C_v \) will have the highest internal energy at 298 K.

From our calculations:
- Ozone (O3) has the highest \( C_v \) value of \( 3 R \).

### Conclusion
Thus, the substance with the maximum internal energy at 298 K is **ozone (O3)**.

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