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To find the number of moles of gas in a sealed container using the ideal gas law, follow these steps:
### Step-by-Step Solution:
1. Identify the Known Quantities and Variables:
- Pressure (P): [tex]\( 1.00 \)[/tex] atm
- Volume (V): [tex]\( 0.325 \)[/tex] liters
- Ideal Gas Constant (R): [tex]\( 0.0821 \)[/tex] L·atm / (mol·K)
- Temperature (T): [tex]\( 293 \)[/tex] Kelvin
2. Write Down the Ideal Gas Law Equation:
The ideal gas law is given by:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure
- [tex]\( V \)[/tex] is the volume
- [tex]\( n \)[/tex] is the number of moles
- [tex]\( R \)[/tex] is the ideal gas constant
- [tex]\( T \)[/tex] is the temperature in Kelvin
3. Rearrange the Equation to Solve for [tex]\( n \)[/tex]:
We want to find [tex]\( n \)[/tex], the number of moles. Rearrange the equation to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
4. Substitute the Known Values into the Equation:
Substitute [tex]\( P \)[/tex], [tex]\( V \)[/tex], [tex]\( R \)[/tex], and [tex]\( T \)[/tex] into the rearranged equation:
[tex]\[ n = \frac{(1.00 \text{ atm}) \times (0.325 \text{ L})}{(0.0821 \text{ L·atm / (mol·K)}) \times (293 \text{ K})} \][/tex]
5. Calculate the Result:
Calculate the numerator and the denominator separately, then divide:
- Numerator: [tex]\( (1.00 \text{ atm}) \times (0.325 \text{ L}) = 0.325 \text{ L·atm} \)[/tex]
- Denominator: [tex]\( (0.0821 \text{ L·atm / (mol·K)}) \times (293 \text{ K}) = 24.0653 \text{ L·atm / mol} \)[/tex]
- Division: [tex]\( \frac{0.325 \text{ L·atm}}{24.0653 \text{ L·atm / mol}} = 0.0135105361396449 \text{ mol} \)[/tex]
Thus, the number of moles of gas that the container can hold is approximately:
[tex]\[ n = 0.0135105361396449 \text{ mol} \][/tex]
This result indicates that the sealed container, under the given conditions of pressure and temperature, can hold around [tex]\( 0.0135 \)[/tex] moles of gas.
### Step-by-Step Solution:
1. Identify the Known Quantities and Variables:
- Pressure (P): [tex]\( 1.00 \)[/tex] atm
- Volume (V): [tex]\( 0.325 \)[/tex] liters
- Ideal Gas Constant (R): [tex]\( 0.0821 \)[/tex] L·atm / (mol·K)
- Temperature (T): [tex]\( 293 \)[/tex] Kelvin
2. Write Down the Ideal Gas Law Equation:
The ideal gas law is given by:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure
- [tex]\( V \)[/tex] is the volume
- [tex]\( n \)[/tex] is the number of moles
- [tex]\( R \)[/tex] is the ideal gas constant
- [tex]\( T \)[/tex] is the temperature in Kelvin
3. Rearrange the Equation to Solve for [tex]\( n \)[/tex]:
We want to find [tex]\( n \)[/tex], the number of moles. Rearrange the equation to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
4. Substitute the Known Values into the Equation:
Substitute [tex]\( P \)[/tex], [tex]\( V \)[/tex], [tex]\( R \)[/tex], and [tex]\( T \)[/tex] into the rearranged equation:
[tex]\[ n = \frac{(1.00 \text{ atm}) \times (0.325 \text{ L})}{(0.0821 \text{ L·atm / (mol·K)}) \times (293 \text{ K})} \][/tex]
5. Calculate the Result:
Calculate the numerator and the denominator separately, then divide:
- Numerator: [tex]\( (1.00 \text{ atm}) \times (0.325 \text{ L}) = 0.325 \text{ L·atm} \)[/tex]
- Denominator: [tex]\( (0.0821 \text{ L·atm / (mol·K)}) \times (293 \text{ K}) = 24.0653 \text{ L·atm / mol} \)[/tex]
- Division: [tex]\( \frac{0.325 \text{ L·atm}}{24.0653 \text{ L·atm / mol}} = 0.0135105361396449 \text{ mol} \)[/tex]
Thus, the number of moles of gas that the container can hold is approximately:
[tex]\[ n = 0.0135105361396449 \text{ mol} \][/tex]
This result indicates that the sealed container, under the given conditions of pressure and temperature, can hold around [tex]\( 0.0135 \)[/tex] moles of gas.
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