The average translational K.E. in one millitre volume of oxygen at NTP is

Options

A. 0.15 J
B. 0.036 J
C. 0.56 J
D. 152 J

Answer

To find the average translational kinetic energy in one milliliter (mL) volume of oxygen gas at Normal Temperature and Pressure (NTP), we can follow these steps:

### Step 1: Understand the properties of oxygen gas
Oxygen is a diatomic gas, which means it has two atoms in each molecule (O2). The degrees of freedom for a diatomic gas are 5: 3 translational and 2 rotational.

### Step 2: Use the formula for average translational kinetic energy
The average translational kinetic energy (K.E.) per molecule can be expressed using the formula:
\[ K.E. = \frac{3}{2} k T \]
where:
- \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \, \text{J/K} \))
- \( T \) is the temperature in Kelvin (at NTP, \( T = 273 \, \text{K} \))

### Step 3: Calculate the average translational kinetic energy per molecule
Substituting the values into the formula:
\[ K.E. = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times (273 \, \text{K}) \]
Calculating this gives:
\[ K.E. = \frac{3}{2} \times 1.38 \times 273 \times 10^{-23} \]
\[ K.E. = \frac{3}{2} \times 3.77 \times 10^{-21} \, \text{J} \]
\[ K.E. = 5.655 \times 10^{-21} \, \text{J} \]

### Step 4: Find the number of molecules in 1 mL of oxygen
At NTP, 1 mole of gas occupies 22.4 liters. Therefore, 1 mL (or \( 1 \times 10^{-3} \) L) contains:
\[ \text{Number of moles} = \frac{1 \times 10^{-3}}{22.4} \approx 4.464 \times 10^{-5} \, \text{moles} \]
Using Avogadro's number (\( 6.022 \times 10^{23} \, \text{molecules/mole} \)), the number of molecules in 1 mL of oxygen is:
\[ \text{Number of molecules} = 4.464 \times 10^{-5} \times 6.022 \times 10^{23} \approx 2.688 \times 10^{19} \, \text{molecules} \]

### Step 5: Calculate the total average translational kinetic energy in 1 mL
The total average translational kinetic energy in 1 mL of oxygen is:
\[ \text{Total K.E.} = \text{Number of molecules} \times K.E. \]
\[ \text{Total K.E.} = 2.688 \times 10^{19} \times 5.655 \times 10^{-21} \, \text{J} \]
Calculating this gives:
\[ \text{Total K.E.} \approx 0.152 \, \text{J} \]

### Final Answer
The average translational kinetic energy in one milliliter volume of oxygen at NTP is approximately **0.15 J**.

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