What is the internal energy of `1.75 ` kg of helium ( atomic mass `= 4.00260 `u ) with a temperature of `100^@ `C ?

Options

A. `4.65 xx 10^3 J`
B. `2.03 xx 10^6 J`
C. `5.44 xx 10^5 J`
D. `8.16 xx 10^6 J`

Answer

To find the internal energy of 1.75 kg of helium gas at a temperature of 100°C, we can use the formula for internal energy of an ideal gas:

\[ U = \frac{f}{2} nRT \]

Where:
- \( U \) = internal energy
- \( f \) = degrees of freedom
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = absolute temperature in Kelvin

### Step 1: Determine the degrees of freedom (f)
For helium, which is a monoatomic gas, the degrees of freedom \( f \) is 3.

### Step 2: Convert the mass of helium to grams
Given mass of helium = 1.75 kg

\[ \text{Mass in grams} = 1.75 \, \text{kg} \times 1000 \, \text{g/kg} = 1750 \, \text{g} \]

### Step 3: Calculate the number of moles (n)
The molecular mass of helium is given as 4.00260 u (where 1 u = 1 g/mol).

\[ \text{Number of moles} (n) = \frac{\text{mass in grams}}{\text{molecular mass}} = \frac{1750 \, \text{g}}{4.00260 \, \text{g/mol}} \approx 437.5 \, \text{mol} \]

### Step 4: Convert the temperature to Kelvin
Given temperature = 100°C

\[ T = 100 + 273 = 373 \, \text{K} \]

### Step 5: Substitute values into the internal energy formula
Now we can substitute \( f \), \( n \), \( R \), and \( T \) into the internal energy formula:

\[ U = \frac{3}{2} nRT = \frac{3}{2} \times 437.5 \, \text{mol} \times 8.314 \, \text{J/(mol·K)} \times 373 \, \text{K} \]

### Step 6: Calculate the internal energy
Calculating the above expression:

\[ U = \frac{3}{2} \times 437.5 \times 8.314 \times 373 \]

Calculating step by step:

1. Calculate \( nRT \):
\[ nRT = 437.5 \times 8.314 \times 373 \approx 1.373 \times 10^6 \, \text{J} \]

2. Now calculate \( U \):
\[ U = \frac{3}{2} \times 1.373 \times 10^6 \approx 2.06 \times 10^6 \, \text{J} \]

Thus, the internal energy \( U \approx 2.06 \times 10^6 \, \text{J} \).

### Final Result
The internal energy of 1.75 kg of helium at 100°C is approximately \( 2.06 \times 10^6 \, \text{J} \).

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