Question

Given each of the following sets of values for three of the gas variables, calculate the unknown quantity. a. \(P=21.2\) atm \(; V=142 \mathrm{~mL} ; n=0.432 \mathrm{~mol} ; T=? \mathrm{~K}\) b. \(P=?\) atm \(; V=1.23 \mathrm{~mL} ; n=0.000115 \mathrm{~mol} ; T=293 \mathrm{~K}\) c. \(P=755 \mathrm{~mm} \mathrm{Hg} ; V=? \mathrm{~mL} ; n=0.473 \mathrm{~mol} ; T=131 \mathrm{C}\)

Short answer

The unknown quantities for each of the given sets are: a. \(T = 466 \mathrm{~K}\) b. \(P = 2.74 \mathrm{~atm}\) c. \(V = 20.27 \mathrm{~L}\)

Step-by-step solution

Identify the unknown variable and given variables

For each set, we will note down the given variables (P, V, n, or T) and the unknown variable we need to find. a. Given - P=21.2 atm, V=142 mL, n=0.432 mol; Unknown - T b. Given - V=1.23 mL, n=0.000115 mol, T=293 K; Unknown - P c. Given - P=755 mmHg, n=0.473 mol, T=131 °C; Unknown - V

Convert units

Before plugging the values into the Ideal Gas Law equation, we must first convert all of the given variables into their appropriate units. For volume, we will convert mL to L by dividing by 1000. For pressure, we will convert mmHg to atm by dividing by 760. For temperature in Celsius, we will convert to Kelvin by adding 273. a. V=142 mL/1000= 0.142 L c. P=755 mmHg/760= 0.9934 atm T=131 °C + 273= 404 K

Plug in the given values and solve for the unknown variable

a. Ideal Gas Law: \( P \times V = n \times R \times T \) Solving for T: \( T = \frac{PV}{nR} \) T = \(\frac{(21.2\text{ atm} \times 0.142\text{ L})}{(0.432\text{ mol} \times 0.0821\frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})}\) T ≈ 466 K b. Ideal Gas Law: \( P \times V = n \times R \times T \) Solving for P: \( P = \frac{n \times R \times T}{V} \) P = \(\frac{(0.000115\text{ mol} \times 0.0821\frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}} \times 293\text{ K})}{(1.23\text{ mL}/1000)}\) P ≈ 2.74 atm c. Ideal Gas Law: \( P \times V = n \times R \times T \) Solving for V: \( V = \frac{n \times R \times T}{P} \) V = \(\frac{(0.473\text{ mol} \times 0.0821\frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}} \times 404\text{ K})}{(0.9934\text{ atm})}\) V ≈ 20.27 L

Final Answers

The unknown quantities for each of the given sets are: a. T = 466 K b. P = 2.74 atm c. V = 20.27 L

Key concepts

Pressure

Pressure is the force exerted by gas particles colliding with the walls of their container. It's an essential concept in understanding gases. Pressure is typically measured in atmospheres (atm) or millimeters of mercury (mmHg).

When working with the Ideal Gas Law, you'll often need to ensure the pressure is in the correct unit (usually atm). If you're given pressure in mmHg, convert it by dividing by 760, since 1 atm equals 760 mmHg. This conversion is crucial for making accurate calculations in gas law problems.

Volume

Volume is the space that a gas occupies. It's usually measured in liters (L) when dealing with the Ideal Gas Law.

If the given volume is in milliliters (mL), convert it to liters by dividing by 1000, as there are 1000 mL in a liter.

This conversion ensures consistency and accuracy in calculations, as the Ideal Gas Law formula requires volume to be in liters. Understanding volume conversion is vital for correctly solving gas-related problems.

Temperature

Temperature reflects the average energy of gas particles and influences how they move and collide. For gas law calculations, temperature must be in Kelvin (K) to work properly within the formula.

To convert from Celsius to Kelvin, simply add 273. This conversion is necessary since the Kelvin scale starts at absolute zero, making it perfect for scientific calculations.

Always double-check that your temperature is in Kelvin when dealing with gas calculations.

Moles of gas

Moles of gas, represented by "n" in formulas, indicate the amount of substance.

The concept of moles allows us to count particles in a straightforward way, using Avogadro's number, which is approximately \(6.022 \times 10^{23}\).

In gas law equations, accurately identifying or calculating the moles is crucial because it links directly to how much gas you're dealing with. This ensures precise measurements and results.

Unit Conversion

Unit conversion is a key step in solving gas law problems. It ensures that all variables are in the correct units to use with the Ideal Gas Law, \( PV = nRT \).

Here are some common conversions:

• Pressure: Convert mmHg to atm by dividing by 760.
• Volume: Convert mL to L by dividing by 1000.
• Temperature: Convert Celsius to Kelvin by adding 273.

Understanding these conversions will help you solve problems accurately, leading to correct and meaningful results.