Which of the following pairs has the same unit?

Options

A. Wavelength and Rydberg constant
B. Relative velocity and relative density
C. Thermal capacity and Boltzmann constant
D. Time period and acceleration gradlient

Explanation

To determine which of the following pairs has the same unit, we will analyze each pair one by one:

1. **Wavelength and Rydberg Constant**:
- The unit of wavelength (λ) is meters (m).
- The Rydberg constant (R) is defined in terms of energy levels in quantum mechanics, and its unit is 1/m (inverse meters).
- Since one is in meters and the other is in inverse meters, they do not have the same unit.

2. **Relative Velocity and Relative Density**:
- Relative velocity is simply a velocity, which has the unit of meters per second (m/s).
- Relative density is a ratio of two densities, which is unitless (it is a comparison of two similar quantities).
- Therefore, relative velocity (m/s) and relative density (unitless) do not have the same unit.

3. **Thermal Capacity and Boltzmann Constant**:
- Thermal capacity (C) is defined as the amount of heat (Q) required to change the temperature (T) of a substance, which has the unit of joules per Kelvin (J/K).
- The Boltzmann constant (k) relates the average kinetic energy of particles in a gas with the temperature and also has the unit of joules per Kelvin (J/K).
- Since both thermal capacity and the Boltzmann constant have the same unit (J/K), this pair is correct.

4. **Time Period and Acceleration Gradient**:
- The time period (T) has the unit of seconds (s).
- The acceleration gradient is the change in acceleration over time, which has the unit of meters per second squared per second (m/s²/s), simplifying to meters per second cubed (m/s³).
- Therefore, time period (s) and acceleration gradient (m/s³) do not have the same unit.

**Conclusion**:
The only pair that has the same unit is **Thermal Capacity and Boltzmann Constant**, both having the unit of joules per Kelvin (J/K).