To determine the molecular formula of the gas, we can follow these steps:
### Step 1: Understand the composition of the gas
The problem states that there are two hydrogen atoms for each carbon atom. We can denote the number of carbon atoms as \( x \). Therefore, the number of hydrogen atoms will be \( 2x \). This gives us the general formula for the gas as \( C_xH_{2x} \).
### Step 2: Use the density of the gas at NTP
The density of the gas at Normal Temperature and Pressure (NTP) is given as \( 1.25 \, \text{g/L} \). At NTP, the pressure is \( 1 \, \text{atm} \) and the temperature is \( 273 \, \text{K} \).
### Step 3: Use the ideal gas equation
The ideal gas equation is given by:
\[
PV = nRT
\]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = gas constant (\( 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \))
- \( T \) = temperature in Kelvin
### Step 4: Relate density to molar mass
We know that the number of moles \( n \) can also be expressed as:
\[
n = \frac{W}{M}
\]
Where:
- \( W \) = weight (mass)
- \( M \) = molar mass
From the density (\( d \)), we can express it as:
\[
d = \frac{W}{V}
\]
Thus, we can rewrite the ideal gas equation in terms of density:
\[
P = \frac{dRT}{M}
\]
### Step 5: Rearranging for molar mass
Rearranging the equation gives us:
\[
M = \frac{dRT}{P}
\]
### Step 6: Substitute the known values
Now, substituting the known values into the equation:
- \( d = 1.25 \, \text{g/L} \)
- \( R = 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \)
- \( T = 273 \, \text{K} \)
- \( P = 1 \, \text{atm} \)
Calculating \( M \):
\[
M = \frac{1.25 \times 0.082 \times 273}{1}
\]
\[
M \approx \frac{27.98}{1} \approx 28 \, \text{g/mol}
\]
### Step 7: Determine the molecular formula
Now we have the molar mass \( M \approx 28 \, \text{g/mol} \).
Using the general formula \( C_xH_{2x} \):
- For \( x = 1 \):
- Molar mass = \( 12 + 2(1) = 14 \, \text{g/mol} \) (not valid)
- For \( x = 2 \):
- Molar mass = \( 12(2) + 2(2) = 24 + 4 = 28 \, \text{g/mol} \) (valid)
Thus, the molecular formula of the gas is \( C_2H_4 \).
### Final Answer:
The molecular formula of the gas is \( C_2H_4 \).
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