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Sagot :
To determine the volume of 6.70 grams of Helium (He) gas at 25.0 °C and a pressure of 740.0 mmHg, we can use the ideal gas law equation, \( PV = nRT \).
Let's break this down step-by-step:
### Step 1: Convert Pressure to Atmospheres
Pressure is given in mmHg, so we need to convert it to atmospheres (atm). The conversion factor is:
[tex]\[ 1 \, \text{atm} = 760 \, \text{mmHg} \][/tex]
Thus, the pressure in atmospheres is:
[tex]\[ \text{Pressure} = \frac{740.0 \, \text{mmHg}}{760.0 \, \text{mmHg/atm}} \approx 0.9737 \, \text{atm} \][/tex]
### Step 2: Convert Temperature to Kelvin
The temperature is given in degrees Celsius, so we need to convert it to Kelvin. The conversion equation is:
[tex]\[ T(\text{K}) = T(^{\circ}\text{C}) + 273.15 \][/tex]
So the temperature in Kelvin is:
[tex]\[ \text{Temperature} = 25.0 + 273.15 = 298.15 \, \text{K} \][/tex]
### Step 3: Calculate the Number of Moles of Helium
To find the number of moles (\( n \)), we use the molar mass of Helium. The molar mass (\( M \)) of Helium (He) is 4.002602 g/mol.
The number of moles is:
[tex]\[ n = \frac{\text{mass}}{\text{molar mass}} \][/tex]
[tex]\[ n = \frac{6.70 \, \text{g}}{4.002602 \, \text{g/mol}} \approx 1.674 \, \text{mol} \][/tex]
### Step 4: Use the Ideal Gas Law to Find the Volume
We have:
[tex]\[ P = 0.9737 \, \text{atm} \][/tex]
[tex]\[ n = 1.674 \, \text{mol} \][/tex]
[tex]\[ R = 0.0821 \, \text{L} \cdot \text{atm}/(\text{K} \cdot \text{mol}) \][/tex]
[tex]\[ T = 298.15 \, \text{K} \][/tex]
The ideal gas law is \( PV = nRT \), and solving for \( V \) (volume) gives:
[tex]\[ V = \frac{nRT}{P} \][/tex]
Substituting the known values:
[tex]\[ V = \frac{1.674 \, \text{mol} \times 0.0821 \, \text{L·atm/(K·mol)} \times 298.15 \, \text{K}}{0.9737 \, \text{atm}} \][/tex]
### Step 5: Calculate the Volume
[tex]\[ V \approx \frac{1.674 \times 0.0821 \times 298.15}{0.9737} \][/tex]
[tex]\[ V \approx 42.08 \, \text{L} \][/tex]
Given the significant figures based on the provided values (3 significant figures), the volume of the Helium gas is:
[tex]\[ \boxed{42.1 \, \text{L}} \][/tex]
Let's break this down step-by-step:
### Step 1: Convert Pressure to Atmospheres
Pressure is given in mmHg, so we need to convert it to atmospheres (atm). The conversion factor is:
[tex]\[ 1 \, \text{atm} = 760 \, \text{mmHg} \][/tex]
Thus, the pressure in atmospheres is:
[tex]\[ \text{Pressure} = \frac{740.0 \, \text{mmHg}}{760.0 \, \text{mmHg/atm}} \approx 0.9737 \, \text{atm} \][/tex]
### Step 2: Convert Temperature to Kelvin
The temperature is given in degrees Celsius, so we need to convert it to Kelvin. The conversion equation is:
[tex]\[ T(\text{K}) = T(^{\circ}\text{C}) + 273.15 \][/tex]
So the temperature in Kelvin is:
[tex]\[ \text{Temperature} = 25.0 + 273.15 = 298.15 \, \text{K} \][/tex]
### Step 3: Calculate the Number of Moles of Helium
To find the number of moles (\( n \)), we use the molar mass of Helium. The molar mass (\( M \)) of Helium (He) is 4.002602 g/mol.
The number of moles is:
[tex]\[ n = \frac{\text{mass}}{\text{molar mass}} \][/tex]
[tex]\[ n = \frac{6.70 \, \text{g}}{4.002602 \, \text{g/mol}} \approx 1.674 \, \text{mol} \][/tex]
### Step 4: Use the Ideal Gas Law to Find the Volume
We have:
[tex]\[ P = 0.9737 \, \text{atm} \][/tex]
[tex]\[ n = 1.674 \, \text{mol} \][/tex]
[tex]\[ R = 0.0821 \, \text{L} \cdot \text{atm}/(\text{K} \cdot \text{mol}) \][/tex]
[tex]\[ T = 298.15 \, \text{K} \][/tex]
The ideal gas law is \( PV = nRT \), and solving for \( V \) (volume) gives:
[tex]\[ V = \frac{nRT}{P} \][/tex]
Substituting the known values:
[tex]\[ V = \frac{1.674 \, \text{mol} \times 0.0821 \, \text{L·atm/(K·mol)} \times 298.15 \, \text{K}}{0.9737 \, \text{atm}} \][/tex]
### Step 5: Calculate the Volume
[tex]\[ V \approx \frac{1.674 \times 0.0821 \times 298.15}{0.9737} \][/tex]
[tex]\[ V \approx 42.08 \, \text{L} \][/tex]
Given the significant figures based on the provided values (3 significant figures), the volume of the Helium gas is:
[tex]\[ \boxed{42.1 \, \text{L}} \][/tex]
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