Answer :
To solve the problem of finding out how many grams of Krypton (Kr) are in a 1.78-liter cylinder at 98.3 °C and 9.40 atm, we can follow these steps:
1. Convert Temperature to Kelvin:
- Start by converting the given temperature from Celsius to Kelvin. You do this because the ideal gas law requires temperature in Kelvin.
- The formula to convert Celsius to Kelvin is:
[tex]\[ \text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15 \][/tex]
- For this problem:
[tex]\[ 98.3 + 273.15 = 371.45 \, \text{K} \][/tex]
2. Use the Ideal Gas Law to Find Moles of Kr:
- The Ideal Gas Law formula is:
[tex]\[ PV = nRT \][/tex]
where [tex]\( P \)[/tex] is pressure in atm, [tex]\( V \)[/tex] is volume in liters, [tex]\( n \)[/tex] is the number of moles, [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L·atm/(K·mol)), and [tex]\( T \)[/tex] is the temperature in Kelvin.
- Rearrange the formula to solve for [tex]\( n \)[/tex] (the number of moles):
[tex]\[ n = \frac{PV}{RT} \][/tex]
- Substitute the known values into the formula:
- [tex]\( P = 9.40 \, \text{atm} \)[/tex]
- [tex]\( V = 1.78 \, \text{L} \)[/tex]
- [tex]\( R = 0.0821 \, \text{L·atm/(K·mol)} \)[/tex]
- [tex]\( T = 371.45 \, \text{K} \)[/tex]
- Calculate:
[tex]\[ n = \frac{9.40 \times 1.78}{0.0821 \times 371.45} \approx 0.5487 \, \text{moles} \][/tex]
3. Calculate the Mass of Kr:
- To find the mass, use the number of moles and the molar mass of Krypton. The molar mass of Kr is approximately 83.798 grams/mole.
- The formula for mass is:
[tex]\[ \text{Mass} = \text{moles} \times \text{molar mass} \][/tex]
- Calculate the mass of Kr:
[tex]\[ \text{Mass} = 0.5487 \times 83.798 \approx 45.98 \, \text{grams} \][/tex]
Therefore, there are approximately 45.98 grams of Krypton in the 1.78-liter cylinder at the given temperature and pressure conditions.
1. Convert Temperature to Kelvin:
- Start by converting the given temperature from Celsius to Kelvin. You do this because the ideal gas law requires temperature in Kelvin.
- The formula to convert Celsius to Kelvin is:
[tex]\[ \text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15 \][/tex]
- For this problem:
[tex]\[ 98.3 + 273.15 = 371.45 \, \text{K} \][/tex]
2. Use the Ideal Gas Law to Find Moles of Kr:
- The Ideal Gas Law formula is:
[tex]\[ PV = nRT \][/tex]
where [tex]\( P \)[/tex] is pressure in atm, [tex]\( V \)[/tex] is volume in liters, [tex]\( n \)[/tex] is the number of moles, [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L·atm/(K·mol)), and [tex]\( T \)[/tex] is the temperature in Kelvin.
- Rearrange the formula to solve for [tex]\( n \)[/tex] (the number of moles):
[tex]\[ n = \frac{PV}{RT} \][/tex]
- Substitute the known values into the formula:
- [tex]\( P = 9.40 \, \text{atm} \)[/tex]
- [tex]\( V = 1.78 \, \text{L} \)[/tex]
- [tex]\( R = 0.0821 \, \text{L·atm/(K·mol)} \)[/tex]
- [tex]\( T = 371.45 \, \text{K} \)[/tex]
- Calculate:
[tex]\[ n = \frac{9.40 \times 1.78}{0.0821 \times 371.45} \approx 0.5487 \, \text{moles} \][/tex]
3. Calculate the Mass of Kr:
- To find the mass, use the number of moles and the molar mass of Krypton. The molar mass of Kr is approximately 83.798 grams/mole.
- The formula for mass is:
[tex]\[ \text{Mass} = \text{moles} \times \text{molar mass} \][/tex]
- Calculate the mass of Kr:
[tex]\[ \text{Mass} = 0.5487 \times 83.798 \approx 45.98 \, \text{grams} \][/tex]
Therefore, there are approximately 45.98 grams of Krypton in the 1.78-liter cylinder at the given temperature and pressure conditions.
