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_2w_2)`
D. `T_2/T_1 = (M_1 w_1)/(M_2 w_2)`

Answer

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

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

### Step 2: Relate Moles to Weight and Molecular Mass
The number of moles \( n \) can be expressed in terms of weight \( w \) and molecular mass \( M \):
\[ n = \frac{w}{M} \]

### Step 3: Substitute Moles in the Ideal Gas Equation
Substituting the expression for \( n \) into the ideal gas law gives:
\[ PV = \frac{w}{M}RT \]

### Step 4: Rearranging for Temperature
We can rearrange this equation to solve for temperature \( T \):
\[ T = \frac{PV \cdot M}{wR} \]

### Step 5: Comparing Two Gases
Now, let’s consider two gases, Gas 1 and Gas 2, with their respective weights \( w_1 \), \( w_2 \) and molecular masses \( M_1 \), \( M_2 \). We can write the equations for both gases:
- For Gas 1:
\[ T_1 = \frac{PV \cdot M_1}{w_1R} \] (Equation A)
- For Gas 2:
\[ T_2 = \frac{PV \cdot M_2}{w_2R} \] (Equation B)

### Step 6: Forming a Ratio of Temperatures
Now, we can form a ratio of the two temperatures:
\[ \frac{T_1}{T_2} = \frac{\frac{PV \cdot M_1}{w_1R}}{\frac{PV \cdot M_2}{w_2R}} \]

### Step 7: Simplifying the Ratio
When we simplify this expression, we see that \( PV \) and \( R \) cancel out:
\[ \frac{T_1}{T_2} = \frac{M_1 \cdot w_2}{w_1 \cdot M_2} \]

### Conclusion
From this derivation, we conclude that the expression:
\[ \frac{T_1}{T_2} = \frac{M_1 \cdot w_2}{w_1 \cdot M_2} \]
is true regarding gas laws, confirming that the first option is correct.

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