Question

A gas starts at the conditions \(78.9 \mathrm{~mL}, 3.008 \mathrm{~atm}\), and \(56^{\circ} \mathrm{C}\). Its conditions change to \(35.6 \mathrm{~mL}\) and \(2.55 \mathrm{~atm}\). What is its final temperature?

Short answer

The final temperature is approximately 126.24 K.

Step-by-step solution

Convert Initial Temperature to Kelvin

The initial temperature is given as \(56^{\circ} \mathrm{C}\). First, convert this to Kelvin by using the formula: \(T(K) = T(\degree C) + 273.15\). So, \(T_1 = 56 + 273.15 = 329.15 \mathrm{~K}\).

Identify Known Variables and Equation

We have an initial volume \(V_1 = 78.9 \mathrm{~mL}\), initial pressure \(P_1 = 3.008 \mathrm{~atm}\), and initial temperature \(T_1 = 329.15 \mathrm{~K}\). The final volume is \(V_2 = 35.6 \mathrm{~mL}\) and final pressure is \(P_2 = 2.55 \mathrm{~atm}\). Use the combined gas law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) to find \(T_2\).

Rearrange the Formula to Find Final Temperature

Rearrange the combined gas law formula to solve for \(T_2\): \(T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}\).

Substitute the Known Values into the Equation

Substitute the known values into the equation: \[T_2 = \frac{2.55 \times 35.6 \times 329.15}{3.008 \times 78.9}\].

Calculate the Final Temperature

Perform the calculation: \[T_2 = \frac{2.55 \times 35.6 \times 329.15}{3.008 \times 78.9} \approx 126.24 \mathrm{~K}\].

Key concepts

Temperature Conversion

The concept of temperature conversion is crucial when working with gas laws, as gases often undergo temperature changes. In the context of the combined gas law, temperature must be in Kelvin to ensure proper calculations. The Kelvin scale is an absolute scale, starting at absolute zero, where there is no thermal motion of molecules.
For converting Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is due to the zero point on the Kelvin scale being at \(-273.15^\circ\mathrm{C}\). The formula used is:
\[ T(K) = T(\degree C) + 273.15 \]
In the original exercise, we started with an initial temperature of\(56^\circ\mathrm{C}\),which converts to
\(329.15\, \mathrm{K}\).It's essential to convert temperatures precisely because any error in conversion can lead to inaccuracy in further calculations.

Gas Laws

Gas laws help us understand how gases behave under different conditions of pressure, volume, and temperature. Several gas laws exist, such as Boyle's Law, Charles's Law, and Avogadro's Law, which, when combined, form the Combined Gas Law. The Combined Gas Law beautifully integrates these individual laws to form:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
This represents how the ratio of pressure and volume to temperature, defined at one state, equates to the other.

• Boyle's Law: At constant temperature, pressure and volume are inversely proportional.
• Charles’s Law: At constant pressure, volume and temperature are directly proportional.
• Avogadro’s Law: Equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.

By understanding how these laws interact in the Combined Gas Law, one can predict the behavior of a gas when it undergoes any change, provided no other variables except for pressure, volume, and temperature change.

Pressure-Volume Relationship

Understanding the relationship between pressure and volume is essential when studying gases. This relationship is captured by Boyle's Law, stating that for a fixed amount of gas at a constant temperature, the product of pressure and volume is a constant. Mathematically, this is expressed as:
\( P_1 V_1 = P_2 V_2 \) (at constant temperature).
In essence, as volume decreases, pressure increases, assuming temperature remains steady and vice versa.
This concept is part of the broader scope of how gases behave under various changes but ties closely to practical applications of the Combined Gas Law. In situations like the exercise, with varying pressures and volumes, we mix this relationship with the understanding that temperature is a factor. Thus, even though pressure and volume might adjust inversely relative to one another, adjustments in a system's temperature can complicate their interactions, making it critical to consider all factors simultaneously in calculations.