Question

Given each of the following sets of values for an ideal gas, calculate the unknown quantity. a. \(P=782 \mathrm{~mm} \mathrm{Hg} ; V=? ; n=0.210 \mathrm{~mol} ; T=27 \quad \mathrm{C}\) b. \(P=? \mathrm{~mm} \mathrm{Hg} ; V=644 \mathrm{~mL} ; n=0.0921 \mathrm{~mol} ; T=303 \mathrm{~K}\) c. \(P=745 \mathrm{~mm} \mathrm{Hg} ; V=11.2 \mathrm{~L} ; n=0.401 \mathrm{~mol} ; T=? \mathrm{~K}\)

Short answer

For the given sets of values for an ideal gas, the calculated unknown quantities are: a. V ≈ 5.33 L b. P ≈ 2706 mmHg c. T ≈ 330 K

Step-by-step solution

Conversion

First, we need to convert the given pressure from mmHg to atm, using the conversion factor: 1 atm = 760 mmHg. Also, we have to convert the given temperature from Celsius to Kelvin using the formula K = ℃ + 273.15. P = \( \frac{782\, mmHg}{760\, mmHg/atm} = 1.03\, atm \) T = 27℃ + 273.15 = 300.15 K (Scene 2: Find the volume V using the Ideal Gas Law)

Find V

Next, we'll plug the given values and the values we just converted into the Ideal Gas Law equation (PV = nRT) and solve for V. 1.03 atm * V = 0.210 mol * 0.0821 L atm/mol K * 300.15 K V = \( \frac{0.210\, mol * 0.0821\, L\, atm/mol\, K * 300.15\, K}{1.03\, atm} \) V ≈ 5.33 L b. P=?, V= 644 mL, n=0.0921 mol, T=303 K → Determine P (Scene 1: Convert volume to L)

Conversion

First, we need to convert the given volume from mL to L, using the conversion factor: 1 L = 1000 mL. V = \( \frac{644\, mL}{1000\, mL/L} = 0.644\, L \) (Scene 2: Find the pressure P using the Ideal Gas Law)

Find P

Next, we'll plug the given values and the value we just converted into the Ideal Gas Law equation (PV = nRT) and solve for P. P * 0.644 L = 0.0921 mol * 0.0821 L atm/mol K * 303 K P = \( \frac{0.0921\, mol * 0.0821\, L\, atm/mol\, K * 303\, K}{0.644\, L} \) P ≈ 3.56 atm To convert P back to mmHg, use the conversion factor: 1 atm = 760 mmHg. P = 3.56 atm * 760 mmHg/atm ≈ 2706 mmHg c. P = 745 mmHg, V = 11.2 L, n= 0.401 mol, T=? → Determine T (Scene 1: Convert pressure to atm)

Conversion

First, we need to convert the given pressure from mmHg to atm, using the conversion factor: 1 atm = 760 mmHg. P = \( \frac{745\, mmHg}{760\, mmHg/atm} = 0.980\, atm \) (Scene 2: Find the temperature T using the Ideal Gas Law)

Find T

Next, we'll plug the given values and the value we just converted into the Ideal Gas Law equation (PV = nRT) and solve for T. 0.980 atm * 11.2 L = 0.401 mol * 0.0821 L atm/mol K * T T = \( \frac{0.980\, atm * 11.2\, L}{0.401\, mol * 0.0821\, L\, atm/mol\, K} \) T ≈ 330 K

Key concepts

Pressure Conversion

When dealing with ideal gases, pressure is often measured in different units, primarily mmHg (millimeters of mercury) or atm (atmospheres). To switch between these units, you need to know the conversion factor: 1 atm is equal to 760 mmHg.
For example, if you're given a pressure of 782 mmHg, you can convert it to atm by using the formula:

• Divide 782 mmHg by 760 mmHg/atm.
• This results in approximately 1.03 atm.

It's crucial to perform this conversion accurately because the Ideal Gas Law typically uses atm for pressure. Always remember to convert pressure to atm before placing the values into the Ideal Gas Law equation.

Temperature Conversion

In problems involving the Ideal Gas Law, temperature needs to be in Kelvin. The Kelvin scale is absolute and relates directly to molecular motion, making it essential for calculation accuracy.
To convert Celsius to Kelvin:

• Add 273.15 to the Celsius temperature.

For instance, if you are given a temperature of 27℃, convert it by adding 273.15:

• This equals 300.15 K.

Always remember to convert the temperature to Kelvin before using it in the Ideal Gas Law, as other temperature scales can lead to incorrect results.

Volume Conversion

Volume measurements in problems can be presented in liters or milliliters. The Ideal Gas Law requires volume in liters, so conversion from milliliters to liters might be necessary.
Use the following method for conversion:

• Divide the volume in milliliters by 1000 to get the volume in liters.

For example, if your volume is 644 mL, divide it by 1000:

• This results in 0.644 L.

Ensure all volumes are in liters before using them in the Ideal Gas Law formula to avoid errors in your calculations.

Calculating Volume

To calculate the unknown volume of an ideal gas, you can rearrange the Ideal Gas Law equation: \[ PV = nRT \]Here, V (volume) can be isolated:\[ V = \frac{nRT}{P} \] This equation means you can find the volume if you know the pressure, temperature, and amount of substance (in mol).
For example, with \( P = 1.03 \) atm, \( n = 0.210 \) mol, \( R = 0.0821 \) L atm/mol K, and \( T = 300.15 \) K, plug in these values:

• Calculate V = \( \frac{0.210 \times 0.0821 \times 300.15}{1.03} \)
• You’ll find V ≈ 5.33 L.

Always ensure you've performed all necessary conversions before this step.

Calculating Pressure

To find the pressure of an ideal gas, use the rearranged version of the Ideal Gas Law:\[ P = \frac{nRT}{V} \]This shows that, given the amount of gas, temperature, and volume, you can solve for pressure.
Imagine you have \( n = 0.0921 \) mol, \( R = 0.0821 \) L atm/mol K, \( T = 303 \) K, and \( V = 0.644 \) L:

• Calculate P = \( \frac{0.0921 \times 0.0821 \times 303}{0.644} \)
• This gives P ≈ 3.56 atm.

After calculating in atm, you might need to convert back to mmHg if required for your final answer.

Calculating Temperature

Determining temperature from the Ideal Gas Law is straightforward by rearranging the equation to solve for temperature:\[ T = \frac{PV}{nR} \]This formula allows you to calculate temperature if you know the pressure, volume, and the amount of gas.
For a situation where \( P = 0.980 \) atm, \( V = 11.2 \) L, \( n = 0.401 \) mol, and \( R = 0.0821 \) L atm/mol K, the calculation proceeds as follows:

• Calculate T = \( \frac{0.980 \times 11.2}{0.401 \times 0.0821} \)
• You'll find T ≈ 330 K.

Always make sure pressure and volume are correctly converted to their respective units before conducting this calculation.