Question

At what temperature, the sample of neon gas would be heated to double its pressure, if the initial volume of gas is reduced by \(15 \%\) at \(75^{\circ} \mathrm{C}\) ? (a) \(319^{\circ} \mathrm{C}\) (b) \(592^{\circ} \mathrm{C}\) (c) \(128^{\circ} \mathrm{C}\) (d) \(60^{\circ} \mathrm{C}\)

Short answer

The temperature at which the sample of neon gas would double its pressure while reducing its volume by 15% at 75°C is (b) 592°C.

Step-by-step solution

Understand the Relationship Between Pressure, Volume, and Temperature

From the ideal gas law, we know that for a fixed amount of gas, the product of pressure (P) and volume (V) is directly proportional to the temperature (T) in kelvins. Mathematically, PV = nRT, where n is the amount of gas in moles and R is the gas constant. Since n and R are constants, if the initial volume decreases by 15%, to keep nR constant, the final P must increase twofold, and the final T must be calculated accordingly.

Calculate the Final Volume of the Gas after the Decrease

The initial reduction in volume is 15%, which means the final volume Vf is 85% of the initial volume Vi, or Vf = 0.85 * Vi. Since we are looking for a relationship rather than absolute values, we can take Vi as 1 unit, and thus Vf will be 0.85 units.

Relate the Initial and Final States Using the Combined Gas Law

The combined gas law is PiVi/Ti = PfVf/Tf, where Pi and Ti are the initial pressure and temperature, and Pf and Tf are the final pressure and temperature. According to the problem, Pf is twice Pi, so Pf = 2Pi. Substituting the volume and pressures, we get Pi * 1 / Ti = 2Pi * 0.85 / Tf.

Simplify the Equation to Solve for Final Temperature Tf

Canceling Pi from both sides and multiplying both sides by Tf to eliminate the fraction leads to Tf / Ti = 2 * 0.85. Next, convert the given initial temperature from Celsius to Kelvin by adding 273.15, to apply the ideal gas law correctly. So Ti becomes 75 + 273.15 Kelvin.

Calculate the Final Temperature Tf in Kelvin

Solving the previous step gives Tf = Ti * 2 * 0.85. Use the converted initial temperature in Kelvin and compute the final temperature in Kelvin.

Convert the Final Temperature Back to Celsius

After finding the final temperature in Kelvin, convert it back to Celsius by subtracting 273.15 to find the temperature at which the pressure of the neon gas doubles.

Key concepts

Combined Gas Law

When dealing with problems involving gas pressures, volumes and temperatures, the Combined Gas Law is an invaluable tool. It simplifies the analysis of gas behavior by merging Charles's Law, Boyle's Law, and Gay-Lussac's Law into one comprehensive equation, represented as \( P_1V_1/T_1 = P_2V_2/T_2 \). This formula allows us to calculate how a gas will respond to changes in pressure (P), volume (V), and temperature (T), assuming the amount of gas remains constant.

The problem from our exercise involves each of these variables. To solve it, we consider the initial state of the gas (with P, V, and T all known values) and relate it to the final state where the volume is decreased by 15% and the pressure has been doubled. Through a rearrangement of the equation, we align the variables accordingly, making it possible to solve for the unknown final temperature. It's crucial when using the Combined Gas Law to always express the temperature in Kelvin to maintain consistency across the calculation.

Temperature-Pressure Relationship

The relationship between temperature and pressure in a gas is a cornerstone of chemical thermodynamics. Gay-Lussac's Law is one piece of this puzzle, stating that at constant volume, the pressure of a gas is directly proportional to its absolute temperature (measured in Kelvin). If we apply this law to a scenario where we double the pressure of a gas, as in our exercise, we expect the temperature to also double provided the volume is kept constant.

In the real-life scenario of our exercise, the volume is not constant—it decreases. This intertwining of pressure and temperature changes requires us to account for the initial conditions (including the initial volume) accurately. As seen in our step-by-step solution, we adjust the equation from the Combined Gas Law to find the final temperature that corresponds to the double pressure condition, after taking into account the reduced volume.

Chemical Thermodynamics

Chemical Thermodynamics deals with the study of energy changes, particularly heat, in chemical reactions and physical transformations. One of its key principles relates to how energy transfers correlate with temperature, volume, and pressure changes in a gas. These relationships help us predict how a gas will behave under different conditions.

For the neon gas in our exercise, understanding thermodynamic principles is vital. The pressure increase upon heating is expected because, according to the first law of thermodynamics, energy cannot be created or destroyed, only transformed. As we provide thermal energy to the gas, it converts this energy into work and internal energy, increasing the temperature, which in turn affects pressure as dictated by the ideal gas law.

Gas Volume Calculations

When performing gas volume calculations, you must always be mindful of the conditions under which a gas is measured. For instance, in our exercise, the volume of gas is reduced by 15%. The volume of a gas is dependent on its pressure and temperature, as outlined by Boyle's Law and Charles's Law, respectively. A decrease or increase in volume can have significant implications for the other variables if temperature or pressure is held constant.

However, in our situation, both pressure and temperature are changing, so we must use the Combined Gas Law to understand these simultaneous effects. By calculating the final volume as a percentage of the initial volume, we set up our problem within the real-world parameters given and find the corresponding temperature at which the gas's pressure doubles. Understanding this interplay between gas properties is crucial when predicting how a gas will behave in a closed system, like many reactions in chemical processes.