Which of the following expression is true regarding gas laws ? ( w = weight , M = molecular mass)

Options

A. `(T_(1))/(T_(2)) = (M_(1) w_2)/(M_2 w_1)`
B. `(T_(1))/(T_(2)) = (M_(2) w_(1))/(M_(1) w_(2))`
C. `(T_(1))/(T_(2)) = (M_(1) w_(1))/(M_2 w_2)`
D. `(T_2)/(T_1) = (M_1 w_1)/(M_2 w_2)`

Explanation

To solve the question regarding which expression is true regarding gas laws, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the Relationship**:
We need to find a relationship between temperature \( T \), weight \( W \), and molecular mass \( M \).

3. **Express Moles in Terms of Weight and Molecular Mass**:
The number of moles \( n \) can be expressed as:
\[ n = \frac{W}{M} \]
where \( W \) is the weight of the gas and \( M \) is the molecular mass.

4. **Substitute into the Ideal Gas Law**:
If we keep pressure \( P \) and volume \( V \) constant, we can express temperature in terms of moles:
\[ T \propto n \quad \text{(at constant P and V)} \]
Thus, we can write:
\[ T \propto \frac{W}{M} \]

5. **Establish the Proportionality**:
From the above relationship, we can say:
\[ T_1 \cdot M_1 = T_2 \cdot M_2 \cdot \frac{W_2}{W_1} \]
Rearranging gives:
\[ \frac{T_1}{T_2} = \frac{W_2 \cdot M_1}{M_2 \cdot W_1} \]

6. **Identify the Correct Expression**:
The final expression derived is:
\[ \frac{T_1}{T_2} = \frac{M_1 \cdot W_2}{M_2 \cdot W_1} \]
This matches with option A.

7. **Conclusion**:
Therefore, the correct expression regarding gas laws is:
\[ \frac{T_1}{T_2} = \frac{M_1 \cdot W_2}{M_2 \cdot W_1} \]
Hence, **Option A is correct**.

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