Question

To what temperature must a sample of nitrogen at 27°C and 0.625 atm be taken so that its pressure becomes 1.125 atm at constant volume?

Short answer

The sample must be taken to 267.12°C for its pressure to become 1.125 atm.

Step-by-step solution

Understand the Exercise

To solve this problem, use the Gay-Lussac's Law which states that at constant volume, the pressure of a gas is directly proportional to its temperature. This can be written as \(\frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature.

Convert the Initial Temperature to Kelvin

The initial temperature is given as 27°C. Convert this to Kelvin by adding 273.15. \(\text{T}_1 = 27 + 273.15 = 300.15 \text{ K}\)

Set Up the Equation

Use Gay-Lussac's Law to set up the equation:\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]Substitute the known values into the equation: \[\frac{0.625 \text{ atm}}{300.15 \text{ K}} = \frac{1.125 \text{ atm}}{T_2}\]

Solve for the Final Temperature

Solve for \(T_2\) by rearranging the equation: \[\frac{T_2}{1.125 \text{ atm}} = \frac{300.15 \text{ K}}{0.625 \text{ atm}}\] \[\text{T}_2 = \frac{300.15 \text{ K}}{0.625} \times 1.125\] Calculate the value: \[\text{T}_2 \approx 540.27 \text{ K}\]

Convert the Final Temperature to Celsius

Convert the final temperature back to Celsius by subtracting 273.15 from the Kelvin temperature. \(\text{T}_2 = 540.27 - 273.15 = 267.12 \text{ °C}\)

Key concepts

Gas Laws

Gas laws describe how gases behave under different conditions of pressure, temperature, and volume. They are fundamental principles in chemistry and physics. Gay-Lussac's Law is one of these important laws. It tells us that, at a constant volume, the pressure of a gas is directly proportional to its temperature. This law is handy for solving problems where you need to find the unknown temperature or pressure of a gas sample. Remember: always make sure the temperature is in Kelvin for gas law calculations. Kelvin is an absolute temperature scale starting from absolute zero.

Pressure-Temperature Relationship

The pressure-temperature relationship is at the core of Gay-Lussac's Law. When the volume of a gas does not change, increasing the temperature will result in an increase in pressure. Conversely, lowering the temperature will decrease the pressure. Imagine a sealed container of gas: if you heat it, the particles move faster and hit the walls harder, raising the pressure. This relationship can be represented by the formula \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\). It is crucial to ensure all units are consistent, especially converting temperatures to Kelvin, since Celsius or Fahrenheit are not suitable for these calculations.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature and their relation to energy and work. The study of thermodynamics includes understanding how thermal energy is converted to and from other forms of energy. In the context of gas laws, we examine how thermal energy affects a gas's behavior in terms of pressure, volume, and temperature. Gay-Lussac's Law fits into the greater framework of thermodynamics since it addresses the changes in internal energy of a gas when temperatures change, assuming volume constant. Remember, in thermodynamics, real-world conditions such as heat loss to surroundings can slightly alter the ideal results predicted by gas laws.

Temperature Conversion

One crucial step in working with gas laws is converting temperature readings to an absolute scale – Kelvin. The Kelvin scale is the standard unit of measurement for thermodynamic temperature in the scientific community. To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, 27°C becomes 300.15 K. Remember, Kelvin does not use the degree symbol (°). This conversion is necessary because the Kelvin scale starts at absolute zero, where theoretically, all molecular motion stops, making it essential for accurate gas law calculations. Using Kelvin ensures proportional relationships in formulas like Gay-Lussac's Law work correctly.