The equation for the speed of sound in a gas states that `v=sqrt(gammak_(B)T//m)`. Speed v is measured in m/s, `gamma` is a dimensionless constant, T is temperature in kelvin (K), and m is mass in kg. Find the SI units for the Boltzmann constant, `k_(B)` ?

Answer

To find the SI units for the Boltzmann constant \( k_B \) from the equation for the speed of sound in a gas, we start with the given equation:

\[ v = \sqrt{\frac{\gamma k_B T}{m}} \]

### Step 1: Identify the units on the left-hand side
The speed \( v \) is measured in meters per second (m/s). Therefore, the left-hand side has the units:

\[ [v] = \text{m/s} \]

### Step 2: Square both sides of the equation
To eliminate the square root, we square both sides:

\[ v^2 = \frac{\gamma k_B T}{m} \]

This gives us:

\[ \left(\frac{\text{m}}{\text{s}}\right)^2 = \frac{\gamma k_B T}{m} \]

### Step 3: Substitute the known units
We know that:
- \( \gamma \) is dimensionless (no units).
- \( T \) (temperature) is measured in Kelvin (K).
- \( m \) (mass) is measured in kilograms (kg).

Substituting these into the equation, we have:

\[ \frac{\text{m}^2}{\text{s}^2} = \frac{k_B \cdot \text{K}}{\text{kg}} \]

### Step 4: Rearrange the equation to isolate \( k_B \)
Rearranging the equation to solve for \( k_B \):

\[ k_B = \frac{\text{m}^2}{\text{s}^2} \cdot \text{kg} \cdot \frac{1}{\text{K}} \]

This simplifies to:

\[ k_B = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2 \cdot \text{K}} \]

### Step 5: Write the final result
Thus, the SI units for the Boltzmann constant \( k_B \) are:

\[ [k_B] = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2 \cdot \text{K}} \]