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To determine the mass of oxygen needed to pressurize a hyperbaric chamber with a volume of 3200 liters to 2.0 atm at a temperature of 21 °C, we can follow these steps.
### Step-by-Step Solution:
1. Convert Temperature to Kelvin:
The temperature in Celsius needs to be converted to Kelvin for use in the Ideal Gas Law calculation.
[tex]\[ \text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15 \][/tex]
So,
[tex]\[ 21 \, ^\circ\text{C} + 273.15 = 294.15 \, \text{K} \][/tex]
2. Apply Ideal Gas Law to Calculate Moles:
The Ideal Gas Law is given by:
[tex]\[ PV = nRT \][/tex]
where:
[tex]\( P \)[/tex] is the pressure in atm (2.0 atm),
[tex]\( V \)[/tex] is the volume in liters (3200 L),
[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 (294.15 K).
Rearrange the equation to solve for the number of moles ([tex]\( n \)[/tex]):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Plugging in the values, we get:
[tex]\[ n = \frac{2.0 \, \text{atm} \times 3200 \, \text{L}}{0.0821 \, \text{L·atm/(K·mol)} \times 294.15 \, \text{K}} \approx 265.01 \, \text{moles} \][/tex]
3. Convert Moles of Oxygen to Grams:
To find the mass of the oxygen, we use the molar mass of oxygen (O₂), which is approximately 32 g/mol.
Calculate the mass:
[tex]\[ \text{Mass of Oxygen} = n \times \text{Molar Mass of Oxygen} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Mass of Oxygen} = 265.01 \, \text{moles} \times 32 \, \text{g/mol} \approx 8480.43 \, \text{grams} \][/tex]
4. Express the Mass to Two Significant Figures:
When rounding to two significant figures, the mass of oxygen is:
[tex]\[ \text{Mass of Oxygen} \approx 8500 \, \text{grams} \][/tex]
### Final Answer:
[tex]\[ \boxed{8500 \, \text{grams}} \][/tex]
Thus, to achieve a pressure of 2.0 atm in a hyperbaric chamber with a volume of 3200 liters at 21 °C, you would need approximately 8500 grams of oxygen.
### Step-by-Step Solution:
1. Convert Temperature to Kelvin:
The temperature in Celsius needs to be converted to Kelvin for use in the Ideal Gas Law calculation.
[tex]\[ \text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15 \][/tex]
So,
[tex]\[ 21 \, ^\circ\text{C} + 273.15 = 294.15 \, \text{K} \][/tex]
2. Apply Ideal Gas Law to Calculate Moles:
The Ideal Gas Law is given by:
[tex]\[ PV = nRT \][/tex]
where:
[tex]\( P \)[/tex] is the pressure in atm (2.0 atm),
[tex]\( V \)[/tex] is the volume in liters (3200 L),
[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 (294.15 K).
Rearrange the equation to solve for the number of moles ([tex]\( n \)[/tex]):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Plugging in the values, we get:
[tex]\[ n = \frac{2.0 \, \text{atm} \times 3200 \, \text{L}}{0.0821 \, \text{L·atm/(K·mol)} \times 294.15 \, \text{K}} \approx 265.01 \, \text{moles} \][/tex]
3. Convert Moles of Oxygen to Grams:
To find the mass of the oxygen, we use the molar mass of oxygen (O₂), which is approximately 32 g/mol.
Calculate the mass:
[tex]\[ \text{Mass of Oxygen} = n \times \text{Molar Mass of Oxygen} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Mass of Oxygen} = 265.01 \, \text{moles} \times 32 \, \text{g/mol} \approx 8480.43 \, \text{grams} \][/tex]
4. Express the Mass to Two Significant Figures:
When rounding to two significant figures, the mass of oxygen is:
[tex]\[ \text{Mass of Oxygen} \approx 8500 \, \text{grams} \][/tex]
### Final Answer:
[tex]\[ \boxed{8500 \, \text{grams}} \][/tex]
Thus, to achieve a pressure of 2.0 atm in a hyperbaric chamber with a volume of 3200 liters at 21 °C, you would need approximately 8500 grams of oxygen.
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