Question

If 1.25 mol of oxygen gas exerts a pressure of \(1200 \mathrm{~mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C}\), what is the volume in liters?

Short answer

The volume of the gas is approximately 19.41 liters.

Step-by-step solution

Identify the Known Variables

We are given several known variables in the problem:- The number of moles of oxygen gas, \( n = 1.25 \text{ mol} \).- The pressure of the gas, \( P = 1200 \text{ mm Hg} \).- The temperature of the gas in Celsius, \( T = 25^{\circ} \text{C} \).

Convert Temperature to Kelvin

Ideal gas law calculations require the temperature to be in Kelvin. Use the conversion formula: \[ T(K) = T(\text{°C}) + 273.15 \]Substitute into the formula:\[ T = 25 + 273.15 = 298.15 \text{ K} \].

Convert Pressure to Atmospheres

The ideal gas law formula \( PV = nRT \) typically uses pressure in atmospheres. Convert \( 1200 \text{ mm Hg} \) to atmospheres using the conversion: \[ 1 \text{ atm} = 760 \text{ mm Hg} \].So, \[ P = \frac{1200}{760} = 1.579 \text{ atm} \].

Use the Ideal Gas Law Formula

The ideal gas law is given by:\[ PV = nRT \]Where:- \( P = 1.579 \text{ atm} \) (pressure)- \( V \) is the volume we need to find.- \( n = 1.25 \text{ mol} \) (number of moles)- \( R = 0.0821 \text{ L atm/mol K} \) (ideal gas constant)- \( T = 298.15 \text{ K} \) (temperature in Kelvin)

Solve for Volume

Rearrange the ideal gas law to solve for volume \( V \):\[ V = \frac{nRT}{P} \]Substitute the known values into the formula:\[ V = \frac{1.25 \times 0.0821 \times 298.15}{1.579} \]Calculate the expression:\[ V = \frac{30.6496875}{1.579} \approx 19.41 \text{ L} \].

Key concepts

Temperature Conversion

When dealing with gas laws, such as the Ideal Gas Law, converting temperature to Kelvin is often necessary. Kelvin is the absolute temperature scale, which starts from absolute zero, where theoretically molecular motion stops. Most gas law calculations require temperature in Kelvin for accurate results.
To convert from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This conversion is quite straightforward. For example:

• If you have a temperature of 25°C, the conversion to Kelvin would be:
• \( T(K) = 25°C + 273.15 = 298.15K \)

Converting temperatures properly ensures your calculations using the Ideal Gas Law (\( PV = nRT \) ) are accurate, as this equation assumes temperatures are measured in Kelvin.

Pressure Conversion

Pressure is an important factor in gas laws, commonly measured in different units including atmospheric pressure (atm), millimeters of mercury (mm Hg), or Pascals (Pa).
In many scientific calculations involving the Ideal Gas Law, it's crucial to convert pressure measurements to atmospheres (\( ext{atm} \) ) since the gas constant \( R \) is often given in terms that include atmospheres.
Converting mm Hg to atm is straightforward:

• Use the conversion factor, \( 1 ext{ atm} = 760 ext{ mm Hg} \) .
• For example, if your pressure is measured at \( 1200 ext{ mm Hg} \), you would convert it by dividing:
• \( P = \frac{1200}{760} = 1.579 ext{ atm} \)

By ensuring pressure units are consistent, you maintain the accuracy of calculations when applying the Ideal Gas Law.

Moles of Gas

The concept of moles is central to the Ideal Gas Law, relating to the quantity of gas present. Moles allow chemists to calculate the number of molecules in a given substance, making them a fundamental unit for expressing amounts of a chemical substance.
The Ideal Gas Law formula, \( PV = nRT \), requires the number of moles (\( n \) ) for accurate calculations. In this scenario, it helps determine how changes in pressure, volume, or temperature affect gas behavior.
Crucial points about moles in gas laws include:

• Moles are a measure of quantity representing a large number of molecules, typically using Avogadro's number (\( 6.022 imes 10^{23} \) molecules/mol).
• For example, if a problem states 1.25 mol of oxygen gas, this figure represents the quantity to be used in the Ideal Gas Law equation.

Understanding the role of moles helps in making accurate predictions about gas conditions under varying environmental factors.