Calculate the internal energy of 1 gram of oxygen at NTP.

Explanation

To calculate the internal energy of 1 gram of oxygen at Normal Temperature and Pressure (NTP), we can follow these steps:

### Step 1: Understand the Conditions
At NTP, the temperature is 273 K (0°C) and the pressure is 1 atm.

### Step 2: Identify the Properties of Oxygen
Oxygen (O2) is a diatomic gas. For a diatomic gas, the degrees of freedom (f) is 5.

### Step 3: Use the Formula for Internal Energy
The internal energy (U) per mole of a diatomic gas can be calculated using the formula:
\[ U = \frac{5}{2} nRT \]
where:
- \( n \) = number of moles
- \( R \) = universal gas constant = 8.31 J/(mol·K)
- \( T \) = temperature in Kelvin

### Step 4: Convert Mass to Moles
First, we need to convert the mass of oxygen (1 gram) to moles. The molecular weight of oxygen (O2) is approximately 32 g/mol. Thus, the number of moles (n) in 1 gram of oxygen is:
\[ n = \frac{\text{mass}}{\text{molecular weight}} = \frac{1 \text{ g}}{32 \text{ g/mol}} = \frac{1}{32} \text{ mol} \]

### Step 5: Substitute Values into the Internal Energy Formula
Now we can substitute the values into the internal energy formula:
\[ U = \frac{5}{2} nRT \]
Substituting \( n = \frac{1}{32} \) mol, \( R = 8.31 \) J/(mol·K), and \( T = 273 \) K:
\[ U = \frac{5}{2} \times \frac{1}{32} \times 8.31 \times 273 \]

### Step 6: Calculate the Internal Energy
Now, we perform the calculation:
1. Calculate \( \frac{5}{2} \times \frac{1}{32} = \frac{5}{64} \)
2. Then, calculate \( 8.31 \times 273 \):
\[ 8.31 \times 273 \approx 2270.43 \text{ J} \]
3. Now, multiply:
\[ U = \frac{5}{64} \times 2270.43 \approx 177.2 \text{ J} \]

### Final Result
The internal energy of 1 gram of oxygen at NTP is approximately **177.2 Joules**.

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