Question

If \(79.0 \mathrm{~L}\) of helium at \(27.0^{\circ} \mathrm{C}\) is compressed to \(32.0 \mathrm{~L}\) at constant pressure, what is the new temperature?

Short answer

So, the new temperature of the helium gas, after compression and maintaining constant pressure, would be approximately 121.5 Kelvin.

Step-by-step solution

Convert the temperatures from Celsius to Kelvin

The initial temperature is given as \(27.0^{\circ} \mathrm{C}\). The conversion from Celsius to Kelvin is done by adding 273.0 to the Celsius value. Thus, \(T1=27.0^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K}\). It's important to note that the temperatures in gas laws must always be in Kelvin.

Set up the Charle's Law equation

According to Charle's Law, the ratio of the initial volume (\( V1 \)) and the initial temperature (\( T1 \)) is equal to the ratio of the final volume (\( V2 \)) and the final temperature (\( T2 \)). The equation becomes \( \frac{V1}{T1} = \frac{V2}{T2} \). Now, substitute all the known values into the equation.

Solve for the new temperature (\( T2 \))

Substituting the known values, we get \( \frac{79.0 \mathrm{L} }{300.15 \mathrm{K}} = \frac{32.0 \mathrm{L}}{T2} \). Solving for \( T2 \), we get \( T2 = \frac{32.0 \mathrm{L} \times 300.15 \mathrm{K}}{79.0 \mathrm{L}} = 121.5 \mathrm{K} \)

Key concepts

Understanding Gas Laws

Gas laws provide an understanding of the relationship between pressure, volume, temperature, and the number of moles in a gas. Charle's Law, specifically, states that the volume of a gas is directly proportional to its temperature when the pressure and the number of moles are held constant. This means that as the temperature of a gas increases, its volume will also increase, provided the gas is in a flexible container. Conversely, if the temperature decreases, so will the volume.

Mainly, Charle's Law can be expressed with the equation \( V1/T1 = V2/T2 \) where \( V \) represents volume and \( T \) represents temperature in Kelvin. Whenever tackling problems related to Charle’s Law, remember to apply this formula to find unknown variables after ensuring all temperatures are expressed in Kelvin, the SI unit for thermodynamic temperature.

The Essentials of Temperature Conversion

When working with gas laws, temperature conversion is vital because these laws require temperatures to be in the Kelvin scale. Celsius can be converted to Kelvin by adding 273.15 to the Celsius value. The formula for this conversion is: \( K = °C + 273.15 \).

It’s important not to confuse Celsius, which is commonly used for everyday temperature measurements, with Kelvin, which is used in scientific calculations to ensure accuracy. It’s also worth noting that Kelvin does not have degrees; it's an absolute scale with zero Kelvin representing absolute zero, the lowest possible thermodynamic temperature.

Deciphering the Kelvin Scale

The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the point at which no more thermal energy can be removed from a system—is 0 K. Kelvin is the base unit of temperature in the International System of Units (SI). Unlike Celsius or Fahrenheit, the Kelvin scale does not use the concept of degrees. A change of 1 K is the same magnitude as a change of 1 degree Celsius, but the Kelvin scale starts at absolute zero rather than water’s freezing point.

Using the Kelvin scale in scientific work, including calculations involving gas laws, allows scientists and students alike to avoid negative temperatures, which simplifies mathematical equations and reflects the true thermodynamic nature of temperature.