Question

If \(1.47 \times 10^{-3} \mathrm{~mol}\) of argon occupies a \(75.0-\mathrm{mL}\) container at \(26^{\circ} \mathrm{C},\) what is the pressure (in torr)?

Short answer

The pressure is 36.6 torr.

Step-by-step solution

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin using the formula: \( T(K) = T(^{\text{∘}}C) + 273.15 \). So, \( T = 26 + 273.15 = 299.15 \text{ K} \).

Convert volume to liters

Convert the volume from milliliters to liters using the relationship: \( 1 \text{ mL } = 10^{-3} \text{ L} \). Hence, \( V = 75.0 \text{ mL } = 75.0 \times 10^{-3} \text{ L} = 0.075 \text{ L} \).

Use the Ideal Gas Law to find pressure

Use the Ideal Gas Law equation: \( PV = nRT \), where: - \( P \) is pressure, - \( V \) is volume, - \( n \) is number of moles, - \( R \) is the ideal gas constant ( \( 0.0821 \text{ L⋅atm/K⋅mol} \) when pressure is in atm), - \( T \) is temperature in Kelvin. Rearrange to solve for pressure: \( P = \frac{nRT}{V} \).

Substitute values

Substitute the known values into the Ideal Gas Law equation: \( n = 1.47 \times 10^{-3} \text{ mol} \), \( R = 0.0821 \text{ L⋅atm/K⋅mol} \), \( T = 299.15 \text{ K } \), \( V = 0.075 \text{ L} \). Thus, \( P = \frac{(1.47 \times 10^{-3} \text{ mol}) (0.0821 \text{ L⋅atm/K⋅mol}) (299.15 \text{ K})}{0.075 \text{ L}} \).

Calculate the pressure in atm

Perform the calculation: \( P = \frac{1.47 \times 10^{-3} \times 0.0821 \times 299.15}{0.075} = 0.0482 \text{ atm} \).

Convert pressure to torr

Convert the pressure from atm to torr using the conversion factor: \( 1 \text{ atm} = 760 \text{ torr} \). Thus, \( P = 0.0482 \text{ atm} \times 760 \text{ torr/atm} = 36.6 \text{ torr} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is an equation that relates the pressure, volume, temperature, and number of moles of a gas using the formula: \( PV = nRT \).

Here's what each symbol stands for:

• P : Pressure of the gas
• V : Volume occupied by the gas
• n : Number of moles of the gas
• R : Ideal gas constant
• T : Temperature of the gas in Kelvin

In our problem, we're given the moles, volume, and temperature so we can use the Ideal Gas Law to find the pressure.
Remember that the value of \( R \) changes based on the units, so be careful to use consistent units in your calculations.

temperature conversion to Kelvin

Temperature must always be in Kelvin when using the Ideal Gas Law.

The conversion formula to switch from Celsius to Kelvin is: \( T(K) = T(^{\text{∘}}C) + 273.15 \).
For example: if the temperature is \( 26^{\circ} \text{C} \), the converted temperature in Kelvin will be: \( 26 + 273.15 = 299.15 \text{ K} \).

This is crucial because using Celsius instead of Kelvin would lead to incorrect calculations. Kelvin makes all temperatures absolute, starting from absolute zero, which is the fundamental basis for the Ideal Gas Law equations.

moles to pressure calculation

To find the pressure of a gas using the Ideal Gas Law, you need to have the number of moles, temperature in Kelvin, volume in liters, and the gas constant.

The equation is rearranged to solve for pressure: \( P = \frac{nRT}{V} \).

Let's substitute the given values for our problem:

• n (number of moles) : \( 1.47 \times 10^{-3} \text{ mol} \)
• R (Ideal Gas Constant) : \( 0.0821 \text{ L⋅atm/K⋅mol} \)
• T (Temperature in Kelvin) : \( 299.15 \text{ K} \)
• V (Volume in Liters) : \( 0.075 \text{ L} \)

Substitute these into the equation: \( P = \frac{(1.47 \times 10^{-3} \text{ mol}) (0.0821 \text{ L⋅atm/K⋅mol}) (299.15 \text{ K})}{0.075 \text{ L}} \).
This gives the pressure in atmospheres (atm), which can then be converted to other units if needed.

volume conversion to liters

The Ideal Gas Law requires volume to be in liters for the units of the gas constant \( R \) to be consistent.

To convert volume from milliliters (mL) to liters (L), use the relationship: \( 1 \text{ mL } = 10^{-3} \text{ L} \).

For example, if we are given a volume of \( 75.0 \text{ mL} \), we convert it to liters by multiplying: \( 75.0 \text{ mL} = 75.0 \times 10^{-3} \text{ L} = 0.075 \text{ L} \).
This step ensures the volume is in the correct units to use in our Ideal Gas Law equation.
Always make sure to double-check unit conversions early to avoid any errors during problem-solving.