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Answer :
To calculate the molecular weight of the gas, we can use the Ideal Gas Law, which states:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure of the gas.
- [tex]\( V \)[/tex] is the volume of the gas.
- [tex]\( n \)[/tex] is the number of moles of the gas.
- [tex]\( R \)[/tex] is the ideal gas constant.
- [tex]\( T \)[/tex] is the temperature in Kelvin.
Given the data:
- Mass of the gas, [tex]\( m = 0.574 \)[/tex] grams
- Volume of the gas, [tex]\( V = 0.548 \times 10^{-3} \)[/tex] cubic meters
- Temperature, [tex]\( T = 22^\circ \text{C} \)[/tex]
- Pressure, [tex]\( P = 98,800 \)[/tex] Newtons per square meter
### Step-by-Step Solution:
1. Convert the temperature to Kelvin:
[tex]\[ T = 22^\circ \text{C} + 273.15 = 295.15 \text{ K} \][/tex]
2. Use the Ideal Gas Law to find the number of moles [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
- [tex]\( P = 98,800 \)[/tex] Nm[tex]\(^2\)[/tex]
- [tex]\( V = 0.548 \times 10^{-3} \)[/tex] m[tex]\(^3\)[/tex]
- [tex]\( R = 8.314 \)[/tex] J/(mol·K)
- [tex]\( T = 295.15 \)[/tex] K
[tex]\[ n = \frac{98,800 \times 0.000548}{8.314 \times 295.15} = 0.022064 \text{ moles} \][/tex]
3. Calculate the molecular weight [tex]\( M \)[/tex]:
The molecular weight is calculated using the formula:
[tex]\[ M = \frac{m}{n} \][/tex]
- [tex]\( m = 0.574 \)[/tex] grams
- [tex]\( n = 0.022064 \)[/tex] moles
[tex]\[ M = \frac{0.574}{0.022064} = 26.015 \text{ g/mol} \][/tex]
### Conclusion:
From the calculations above, the molecular weight of the gas is approximately [tex]\( 26.1 \)[/tex] g/mol.
Therefore, the correct answer is:
C. 26.1
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure of the gas.
- [tex]\( V \)[/tex] is the volume of the gas.
- [tex]\( n \)[/tex] is the number of moles of the gas.
- [tex]\( R \)[/tex] is the ideal gas constant.
- [tex]\( T \)[/tex] is the temperature in Kelvin.
Given the data:
- Mass of the gas, [tex]\( m = 0.574 \)[/tex] grams
- Volume of the gas, [tex]\( V = 0.548 \times 10^{-3} \)[/tex] cubic meters
- Temperature, [tex]\( T = 22^\circ \text{C} \)[/tex]
- Pressure, [tex]\( P = 98,800 \)[/tex] Newtons per square meter
### Step-by-Step Solution:
1. Convert the temperature to Kelvin:
[tex]\[ T = 22^\circ \text{C} + 273.15 = 295.15 \text{ K} \][/tex]
2. Use the Ideal Gas Law to find the number of moles [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
- [tex]\( P = 98,800 \)[/tex] Nm[tex]\(^2\)[/tex]
- [tex]\( V = 0.548 \times 10^{-3} \)[/tex] m[tex]\(^3\)[/tex]
- [tex]\( R = 8.314 \)[/tex] J/(mol·K)
- [tex]\( T = 295.15 \)[/tex] K
[tex]\[ n = \frac{98,800 \times 0.000548}{8.314 \times 295.15} = 0.022064 \text{ moles} \][/tex]
3. Calculate the molecular weight [tex]\( M \)[/tex]:
The molecular weight is calculated using the formula:
[tex]\[ M = \frac{m}{n} \][/tex]
- [tex]\( m = 0.574 \)[/tex] grams
- [tex]\( n = 0.022064 \)[/tex] moles
[tex]\[ M = \frac{0.574}{0.022064} = 26.015 \text{ g/mol} \][/tex]
### Conclusion:
From the calculations above, the molecular weight of the gas is approximately [tex]\( 26.1 \)[/tex] g/mol.
Therefore, the correct answer is:
C. 26.1
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