Problem 13 - Chemistry in 30 days

Question

If pressure of a gas contained in a closed vessel is increased by \(0.4 \%\) when heated by \(1^{\circ} \mathrm{C}\). its initial temperature must be (a) \(250 \mathrm{~K}\) (b) \(250^{\circ} \mathrm{C}\) (c) \(2500 \mathrm{~K}\) (d) \(25^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The initial temperature is (a) 250 K.

Step 1: Understanding the Problem

We are given that the pressure of a gas in a closed vessel increases by 0.4% when the temperature is increased by 1°C. We need to find the initial temperature of the gas.

Step 2: Relate Pressure and Temperature

In a closed system, according to Gay-Lussac's law, the pressure of a gas is directly proportional to its temperature when the volume is constant, i.e., \(\frac{P}{T} = \text{constant}\). Here, \(P\) is pressure and \(T\) is temperature.

Step 3: Express the Change mathematically

Given the pressure change is 0.4%, we can write the final pressure as: \(P_f = P_i + 0.004 P_i = 1.004 P_i\). Since the temperature increases by 1°C, the final temperature \(T_f = T_i + 1\).

Step 4: Set up the Proportion

Applying Gay-Lussac's law, we set up the proportion: \(\frac{P_f}{T_f} = \frac{P_i}{T_i}\), which implies \(\frac{1.004 P_i}{T_i + 1} = \frac{P_i}{T_i}\).

Step 5: Solve the Equation

Cancel \(P_i\) from both sides and solve: \(1.004 = \frac{T_i}{T_i + 1}\). Rearrange to find: \(1.004(T_i + 1) = T_i\).

Step 6: Simplify and Solve for Initial Temperature

Expand and simplify the equation: \(1.004 T_i + 1.004 = T_i\). Rearrange terms to find \(T_i = 1.004/0.004\). Calculate \(T_i\).

Step 7: Calculate the Initial Temperature

Calculate \(T_i\): \(T_i = 251\), considering unit conversion we see the answer should be in Kelvin.

Key Concepts

Pressure-Temperature Relationship

In the study of gases, understanding the connection between pressure and temperature is vital. This relationship is famously described by Gay-Lussac's law, which states that the pressure of a gas is directly proportional to its temperature when the volume is held constant. This can be expressed through the formula: \( \frac{P}{T} = \text{constant} \). Here, \( P \) represents pressure and \( T \) signifies temperature.

In practical terms, this means:

• If the temperature of a closed system gas increases, its pressure also increases, provided the volume doesn't change.
• Conversely, if the temperature decreases, the pressure decreases as well.

When analyzing changes, it's important to engage with the percentage change in pressure or temperature, interpreting how a small shift in one can affect the other in a proportional manner.

Closed System Gas Behavior

A closed system means the gas is sealed within a container where no particles can enter or leave. This isolation ensures the system remains unchanged externally, making it ideal for observing how internal conditions like pressure and temperature interact.

In such a system:

• The volume of the gas doesn't change, which is a key condition for applying Gay-Lussac's law.
• Energy changes occur in the form of heat transfer affecting temperature and pressure.

Understanding closed systems helps in making predictions about how gases will behave under various thermal conditions. Such an isolated environment provides a controlled basis for validating theoretical gas laws in real-life scenarios.

Kelvin to Celsius Conversion

Temperature measurements in scientific settings often require conversion between two units: Kelvin and Celsius. Understanding these conversions is critical to accurately interpreting scientific results. The Kelvin scale is based on absolute zero, the point at which no thermal energy remains in a system.

To convert between these:

• The formula for Celsius to Kelvin is \( K = °C + 273.15 \).
• For Kelvin back to Celsius, use \( °C = K - 273.15 \).

In our original problem, after solving the equation to find an initial temperature of 251, it is key to address whether this temperature was in Kelvin or Celsius. Since gas laws typically use Kelvin for absolute temperature reference, interpreting results carefully ensures accuracy in solutions.