Chapter 13: Problem 30
Challenge An ideal gas has a volume of 3.0 L. If the number of moles of gas and the temperature are doubled, while the pressure remains constant, what is the new volume?
Short Answer
Expert verified
The new volume of the ideal gas, after doubling the number of moles and the temperature while keeping the pressure constant, is 12.0 L.
Step by step solution
01
Write the ideal gas law formula and identify the variables
The ideal gas law formula is given by PV = nRT, where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the ideal gas constant (8.314 J/molK)
- T is the temperature of the gas
In this problem, we are to find the new volume V₂, given that n and T are doubled, and P remains constant.
02
Set up a ratio comparing initial and final states
Since the only variable that remains constant during the process is pressure, we can set up a ratio comparing the initial and final states of the gas:
\( \frac{V_1}{n_1T_1} = \frac{V_2}{n_2T_2} \)
03
Use the given information to fill in the known values
We are given that the initial volume is V₁ = 3.0 L, the number of moles and temperature are both doubled, and the pressure remains constant. We can plug this into our ratio:
\( \frac{\text{3.0 L}}{n_1T_1} = \frac{V_2}{\text{2}n_1(\text{2}T_1)} \)
04
Simplify and solve for V₂
Now, we can simplify the equation by canceling the values of n₁ and T₁:
\( \frac{\text{3.0 L}}{1} = \frac{V_2}{\text{4}} \)
Next, solve for the new volume, V₂.
\( V_2 = 3.0\,\text{L} \times 4 \)
\( V_2 = 12.0\,\text{L} \)
05
State the final answer
The new volume of the ideal gas, after doubling the number of moles and the temperature while keeping the pressure constant, is 12.0 L.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volume Change
When discussing an ideal gas, one of the key aspects you might encounter is volume change. Essentially, volume refers to the amount of space that a substance occupies. In gases, this can change based on conditions such as temperature, pressure, or the quantity of the gas itself. In our problem, both the number of moles and the temperature double while the pressure stays constant. Hence, according to the formula variation of the ideal gas law, this leads to a change in volume.
The relationship between these variables is quite straightforward:
- If the amount of gas (moles) increases, the volume tends to increase proportionally, assuming the pressure and temperature are unchanged.
- If the temperature increases, at constant pressure and same amount of gas, the volume also tends to increase.
Number of Moles
The number of moles is a fundamental unit in chemistry that provides a count of the particles in a sample. When we talk about gases, the number of moles directly influences the volume the gas occupies, given a set pressure and temperature.
For instance:
- Doubling the moles of a gas causes the volume to also double, assuming pressure and temperature remain fixed.
- More moles equate to more particles, which naturally requires more space for the particles to move around in a gaseous state.
Temperature
Temperature is a measure of how hot or cold something is and is closely related to the energy of the particles in a gas. In the case of an ideal gas, temperature is directly proportional to the kinetic energy of the gas's molecules.
Here's how temperature affects the behavior of a gas:
- Increasing the temperature generally increases the average energy of the molecules, resulting in greater movement.
- This increased movement leads to an expansion in the gas’s volume if the pressure is kept constant.
Constant Pressure
Constant pressure is an important condition often considered in problems related to gases. It means the pressure within the gas is maintained as other properties change, such as volume or temperature. In the ideal gas law, if the pressure remains constant, adjustments in the number of moles or temperature will lead to corresponding changes in the volume.
Constant pressure can be represented in various situations:
- In experiments where volume or temperature is modified, maintaining a constant pressure allows scientists to observe pure effects of one variable on another.
- Real-life applications like balloons or piston mechanisms, where pressure remains constant allow for the study of volume and temperature relationships.