Question

One $1.00\mathrm{mol}$ of an ideal gas is held at a constant volume of $2.00\mathrm{L}$. Find the change in pressure if the temperature increases by $100.{}^{\circ }\mathrm{C}$.

Short answer

Title: Change in Pressure of an Ideal Gas Question: Given that the amount of an ideal gas is 1.00 mol, the volume is held constant at 2.00 L, and the temperature increases by 100°C, find the change in pressure (ΔP). Answer: ΔP = Pi((Ti + ΔT) / Ti - 1) Explanation: We used the ideal gas law equation and the given information to create an equation for the change in pressure as a function of the initial pressure and temperature. While we cannot find the specific change in pressure without knowing the initial temperature, this equation can be used for any initial temperature to find the change in pressure when the temperature increases by 100 °C.

Step-by-step solution

Convert given temperature to Kelvin

First, we need to convert the increase in temperature from Celsius to Kelvin. To do this, simply add 273.15 to the given value: $\mathrm{\Delta }T=100°C+273.15=373.15\mathrm{K}$ This will be the new temperature when the increase is added to the original temperature.

Find initial and final temperatures

Since we don't have the initial temperature, let's assume that $T_i$ is the initial temperature and $T_f$ is the final temperature. Thus: $T_f=T_i+\mathrm{\Delta }T$

Calculate initial and final pressures

Now, we will use the ideal gas law to find the initial and final pressures: Initial pressure: $P_{i}V=nR{T}_i$ Final pressure: $P_{f}V=nR{T}_f$

Find the change in pressure

To find the change in pressure, you can set up the following equation: $\mathrm{\Delta }P=P_f-P_i$ From Step 2, we know that $T_f=T_i+\mathrm{\Delta }T$. We can substitute that into the equation for final pressure: $P_{f}V=nR(T_i+\mathrm{\Delta }T)$ Now, divide the initial pressure equation by the final pressure equation: $\frac{P_i}{P_f}=\frac{nR{T}_i}{nR(T_i+\mathrm{\Delta }T)}$ Simplify the equation: $\frac{P_i}{P_f}=\frac{T_i}{T_i+\mathrm{\Delta }T}$ Now, solving for $P_f$, we get: $P_f=P_i\frac{T_i+\mathrm{\Delta }T}{T_i}$ Substituting this back into the equation for the change in pressure: $\mathrm{\Delta }P=P_i\frac{T_i+\mathrm{\Delta }T}{T_i}-P_i$ Factor out $P_i$: $\mathrm{\Delta }P=P_i(\frac{T_i+\mathrm{\Delta }T}{T_i}-1)$ This provides us with the change in pressure as a function of the initial pressure and temperature. Unfortunately, we cannot find the specific change in pressure without knowing the initial temperature, but this equation can be used for any initial temperature to find the change in pressure when the temperature increases by 100 °C.

Key concepts

Temperature change in Kelvin

Understanding the temperature change in Kelvin is crucial when working with the Ideal Gas Law as it ensures uniformity in measurements, considering temperature is a fundamental part of this law. The Kelvin scale is an absolute temperature scale starting from 'absolute zero', the point at which particles have minimal kinetic energy.

In our exercise, the given temperature increase is 100 °C, which needs to be converted into Kelvin because the Ideal Gas Law requires temperature to be measured in this scale. To convert Celsius to Kelvin, 273.15 is added to the Celsius value. Hence, an increase of $100°C$ converts to an increase of $373.15K$.

It's important to recognize that a temperature change in Kelvin is equivalent in magnitude to a change in Celsius; the numerical difference does not change. However, because the scales have different starting points, their absolute readings are not the same. This step is vital to avoid any miscalculation when you apply this to the Ideal Gas Law.

Initial and final pressures

Identifying initial and final pressures of a gas system is a key part of solving many thermodynamics problems. In our scenario with the ideal gas, we are looking for the difference in pressure as a result of a temperature change, at constant volume.

Using the Ideal Gas Law $P=\frac{nRT}{V}$ where $P$ is pressure, $n$ is moles of gas, $R$ is the universal gas constant, $T$ is temperature, and $V$ is volume, we can correlate pressure with temperature given that the amount of gas and the volume remain constant.

The initial pressure $P_i$ is what we have before the temperature change and the final pressure $P_f$ is the pressure after the temperature rises. They are calculated by substituting the initial and final temperatures (measured in Kelvin) into the Ideal Gas Law. It's critical to notate that the equation assumes a closed system where no gas can enter or exit.

Change in pressure calculation

The calculation of the change in pressure once the temperature changes is vital in predicting how gases will behave under different conditions. Once we have our initial pressure $P_i$ and the temperature has been increased, we can use the formula $\mathrm{\Delta }P=P_f-P_i$ to find the change in pressure.

By rearranging and substituting the temperature increment into the Ideal Gas Law, as shown in the provided exercise solution, we can express the final pressure in terms of the initial pressure and temperature. Doing so allows us to derive a formula solely dependent on these initial states, making it versatile for multiple scenarios.

The final expression $\mathrm{\Delta }P=P_i(\frac{T_i+\mathrm{\Delta }T}{T_i}-1)$ illuminates how both an increase in temperature $\mathrm{\Delta }T$ and the initial temperature $T_i$ work together to affect the pressure change. In practice, had we known $T_i$ in our exercise, we could plug in these values to find the exact change in pressure $\mathrm{\Delta }P$ after the temperature increase of 100 °C. Without that, we establish a relationship that can apply to any initial value of temperature.