Ideal gas law equation is applied in the following problems: 6. How many grams of helium are required to fill a $$ 4-\mathrm{L} $$ balloon to a pressure of 1.1 atm at $$ 25^{\circ} \mathrm{C} $$ ? 7. 22.5 kg of helium are placed in a balloon. What is the volume ( L ) of the balloon if the final pressure is 1.20 atm and the temperature is $$ 22^{\circ} \mathrm{C} $$ ? 8. A 1 gram sample of water is allowed to vaporize completely in a 10 -L container. What is the pressure of the water vapor at a temperature of $$ 150^{\circ} \mathrm{C} $$ ? 9. You have placed 5.5 g of hexane, $$ \mathrm{C}_{6} \mathrm{H}_{14} $$, in the cylinder of an automobile engine. What is the pressure of the hexane vapor if the cylinder has a volume of $$ 250 \mathrm{~cm}^{3} $$ and the temperature is $$ 250^{\circ} \mathrm{C} $$ ? 10. The average outdoor concentration of carbon monoxide, CO , is about $$ 1000 \mu \mathrm{~g} \mathrm{~m} $$ is this concentration expressed on the ppm and ppb scale? Assume that the outdoor temperature is $$ 17^{\circ} \mathrm{C} $$ and that the total air pressure is 1.04 atm .

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To solve these problems, we will use the Ideal Gas Law, which is given by the equation:

$$
PV = nRT
$$

Where:
- $$ P $$ = pressure (in atm)
- $$ V $$ = volume (in liters)
- $$ n $$ = number of moles of gas
- $$ R $$ = ideal gas constant $$ (0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol}) $$
- $$ T $$ = temperature (in Kelvin)

We will also need to convert temperatures from Celsius to Kelvin using the formula:

$$
T(K) = T(°C) + 273.15
$$

Let's solve each problem step by step.

### Problem 6
**How many grams of helium are required to fill a $$ 4-\mathrm{L} $$ balloon to a pressure of 1.1 atm at $$ 25^{\circ} \mathrm{C} $$?**

1. Convert temperature to Kelvin:
   $$
   T = 25 + 273.15 = 298.15 \, K
   $$

2. Use the Ideal Gas Law to find $$ n $$:
   $$
   n = \frac{PV}{RT}
   $$
   Substituting the values:
   $$
   n = \frac{(1.1 \, \text{atm})(4 \, \text{L})}{(0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol})(298.15 \, K)}
   $$

3. Calculate $$ n $$.

4. Convert moles of helium to grams using the molar mass of helium ($$ 4.00 \, \text{g/mol} $$):
   $$
   \text{grams} = n \times 4.00
   $$

### Problem 7
**22.5 kg of helium are placed in a balloon. What is the volume (L) of the balloon if the final pressure is 1.20 atm and the temperature is $$ 22^{\circ} \mathrm{C} $$?**

1. Convert mass to moles:
   $$
   n = \frac{22.5 \, \text{kg}}{0.004 \, \text{kg/mol}} = 5625 \, \text{mol}
   $$

2. Convert temperature to Kelvin:
   $$
   T = 22 + 273.15 = 295.15 \, K
   $$

3. Use the Ideal Gas Law to find $$ V $$:
   $$
   V = \frac{nRT}{P}
   $$
   Substituting the values:
   $$
   V = \frac{(5625 \, \text{mol})(0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol})(295.15 \, K)}{1.20 \, \text{atm}}
   $$

### Problem 8
**A 1 gram sample of water is allowed to vaporize completely in a 10-L container. What is the pressure of the water vapor at a temperature of $$ 150^{\circ} \mathrm{C} $$?**

1. Convert temperature to Kelvin:
   $$
   T = 150 + 273.15 = 423.15 \, K
   $$

2. Convert mass to moles of water:
   $$
   n = \frac{1 \, \text{g}}{18.015 \, \text{g/mol}} \approx 0.0555 \, \text{mol}
   $$

3. Use the Ideal Gas Law to find $$ P $$:
   $$
   P = \frac{nRT}{V}
   $$
   Substituting the values:
   $$
   P = \frac{(0.0555 \, \text{mol})(0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol})(423.15 \, K)}{10 \, \text{L}}
   $$

### Problem 9
**You have placed 5.5 g of hexane, $$ \mathrm{C}_{6} \mathrm{H}_{14} $$, in the cylinder of an automobile engine. What is the pressure of the hexane vapor if the cylinder has a volume of $$ 250 \mathrm{~cm}^{3} $$ and the temperature is $$ 250^{\circ} \mathrm{C} $$?**

1. Convert volume to liters:
   $$
   V = 250 \, \text{cm}^3 = 0.250 \, \text{L}
   $$

2. Convert temperature to Kelvin:
   $$
   T = 250 + 273.15 = 523.15 \, K
   $$

3. Convert mass to moles of hexane:
   $$
   n = \frac{5.5 \, \text{g}}{86.18 \, \text{g/mol}} \approx 0.0638 \, \text{mol}
   $$

4. Use the Ideal Gas Law to find $$ P $$:
   $$
   P = \frac{nRT}{V}
   $$
   Substituting the values:
   $$
   P = \frac{(0.0638 \, \text{mol})(0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol})(523.15 \, K)}{0.250 \, \text{L}}
   $$

### Problem 10
**The average outdoor concentration of carbon monoxide, CO, is about $$ 1000 \mu \mathrm{g} \mathrm{~m} $$. Is this concentration expressed on the ppm and ppb scale? Assume that the outdoor temperature is $$ 17^{\circ} \mathrm{C} $$ and that the total air pressure is 1.04 atm.**

1. Convert temperature to Kelvin:
   $$
   T = 17 + 273.15 = 290.15 \, K
   $$

2. Calculate the number of moles of air in 1 m³ (1000 L):
   $$
   n = \frac{PV}{RT}
   $$
   Substituting the values:
   $$
   n = \frac{(1.04 \, \text{atm})(1000 \, \text{L})}{(0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol})(290.15 \, K)}
   $$

3. Convert $$ 1000 \mu \mathrm{g
To solve these problems, use the Ideal Gas Law $$ PV = nRT $$. Convert temperatures to Kelvin, calculate moles for each gas, and apply the formula to find the required values.

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