The unit of universal gas constant is

Options

A. `"watt"//"K"`
B. `"dyne"//^(circ)"C"`
C. `"erg"//"K"`
D. `"newton"//^(circ)"R"`

Explanation

To find the unit of the universal gas constant (R), we can start from the ideal gas equation:

1. **Write the Ideal Gas Equation**:
The ideal gas equation is given by:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Rearrange the Equation**:
We can rearrange the equation to solve for \( R \):
\[ R = \frac{PV}{nT} \]

3. **Identify the Units of Each Variable**:
- The unit of pressure \( P \) is typically measured in pascals (Pa), which is equivalent to \( \text{N/m}^2 \) (newtons per square meter).
- The unit of volume \( V \) is measured in cubic meters (m³).
- The unit of temperature \( T \) is measured in Kelvin (K).
- The unit of the number of moles \( n \) is measured in moles (mol).

4. **Substitute the Units into the Equation**:
Now, substituting the units into the equation for \( R \):
\[ R = \frac{(P \text{ in Pa})(V \text{ in m}^3)}{(n \text{ in mol})(T \text{ in K})} \]
\[ R = \frac{(\text{N/m}^2)(\text{m}^3)}{(\text{mol})(\text{K})} \]

5. **Simplify the Units**:
We know that \( \text{N} = \text{kg} \cdot \text{m/s}^2 \), thus:
\[ R = \frac{(\text{kg} \cdot \text{m/s}^2)(\text{m}^3)}{(\text{mol})(\text{K})} \]
\[ R = \frac{\text{kg} \cdot \text{m}^4}{\text{s}^2 \cdot \text{mol} \cdot \text{K}} \]

6. **Identify the Common Units for R**:
In terms of commonly used units, we can express \( R \) in terms of energy per temperature:
Since \( \text{kg} \cdot \text{m}^2/\text{s}^2 \) is equivalent to Joules (J), we can say:
\[ R = \frac{\text{J}}{\text{mol} \cdot \text{K}} \]

7. **Final Unit of Universal Gas Constant**:
Therefore, the unit of the universal gas constant \( R \) can also be expressed as:
\[ R = \frac{\text{J}}{\text{mol} \cdot \text{K}} \]
In terms of the options given, Joules can also be represented as \( \text{L} \cdot \text{atm} \) or \( \text{m}^3 \cdot \text{Pa} \).

8. **Conclusion**:
Among the given options, the unit of the universal gas constant is:
\[ \text{Option 3: J/(mol \cdot K)} \] or in terms of the options provided, it corresponds to \( \text{arc/K} \).