Study of the variation in volume with pressure for a sample of air at constant temperature
There was a problem loading the mathematical equations. Please refresh the page.
1. Aim
To study the variation in volume with pressure for a sample of air at constant temperature by plotting graphs between P and V, and between P and 1/V, and to verify Boyle's Law.
2. Apparatus Used
- Boyle's Law apparatus consisting of a pressure gauge and a graduated cylinder
- Mercury
- A thermometer
- Graph paper
- Ruler
- Calculator
- Barometer (to measure atmospheric pressure)
3. Diagram
Fig. 1: Experimental setup for studying Boyle's Law
4. Theory
Boyle's Law states that at constant temperature, the pressure of a given mass of gas is inversely proportional to its volume. This can be mathematically expressed as:
\[ P \propto \frac{1}{V} \]
\[ PV = \text{constant} \]
Where:
- P = Pressure of the gas
- V = Volume of the gas
This means that if the pressure increases, the volume must decrease proportionally to maintain the constant product. Similarly, if the volume increases, the pressure must decrease.
According to the kinetic theory of gases, this relationship can be explained as follows:
- When the volume decreases, gas molecules are forced into a smaller space
- This causes an increase in the number of collisions of molecules with the walls of the container
- The increased collision frequency results in greater pressure
The graphical representation of Boyle's Law shows:
- A hyperbolic curve when pressure (P) is plotted against volume (V)
- A straight line passing through the origin when pressure (P) is plotted against reciprocal of volume (1/V)
5. Formula
The mathematical expression of Boyle's Law is:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \(P_1\) and \(V_1\) are the initial pressure and volume
- \(P_2\) and \(V_2\) are the final pressure and volume
For verification, we can write:
\[ P \times V = k \]
Where k is a constant for a fixed mass of gas at constant temperature.
From this, we can derive:
\[ P = k \times \frac{1}{V} \]
This shows that P is directly proportional to 1/V, which should give a straight line when plotted.
6. Procedure
- Set up the Boyle's Law apparatus as shown in the diagram.
- Ensure that the temperature of the laboratory remains constant throughout the experiment.
- Record the room temperature using the thermometer.
- Measure the atmospheric pressure using the barometer.
- Initially set the apparatus to have a reasonable volume of trapped air in the graduated cylinder.
- Record the initial volume (V) and pressure (P) readings.
- Apply pressure to the air column by adjusting the height of the mercury column.
- Record the new pressure and corresponding volume.
- Repeat steps 7-8 for at least 8-10 different pressure values.
- Calculate 1/V for each volume reading.
- Calculate the product P × V for each set of readings.
- Plot a graph of P versus V (pressure on y-axis, volume on x-axis).
- Plot another graph of P versus 1/V (pressure on y-axis, 1/V on x-axis).
- Analyze the graphs to verify Boyle's Law.
7. Observation Table
Room Temperature: ________ °C
Atmospheric Pressure: ________ mm Hg
| S.No. | Pressure (P) (N/m²) |
Volume (V) (m³) |
1/V (m⁻³) |
P × V (N·m) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 | ||||
| 7 | ||||
| 8 |
8. Calculations
For each set of readings:
- Calculate 1/V by taking the reciprocal of the volume reading.
- Calculate P × V by multiplying the pressure and volume values.
For the graph of P vs 1/V:
\[ P = k \times \frac{1}{V} \]
The slope of this line gives the value of the constant k, which should be approximately equal to the calculated P × V values.
Sample calculation for one set of readings:
\begin{align}
\text{Given:} \\
\text{Pressure (P)} &= 1.5 \times 10^5 \ \text{N/m}^2 \\
\text{Volume (V)} &= 2.0 \times 10^{-3} \ \text{m}^3 \\
\\
\text{Calculate 1/V:} \\
\frac{1}{V} &= \frac{1}{2.0 \times 10^{-3}} = 5.0 \times 10^2 \ \text{m}^{-3} \\
\\
\text{Calculate P} \times \text{V:} \\
P \times V &= 1.5 \times 10^5 \times 2.0 \times 10^{-3} \\
&= 3.0 \times 10^2 \ \text{N} \cdot \text{m} \text{ or Joules}
\end{align}
9. Result
- The graph between P and V shows a hyperbolic curve, demonstrating an inverse relationship between pressure and volume.
- The graph between P and 1/V shows a straight line passing through the origin, confirming that pressure is directly proportional to the reciprocal of volume.
- The value of the constant k (P × V) is found to be __________ N·m (Joules).
- The experimental results verify Boyle's Law that at constant temperature, the pressure of a given mass of gas is inversely proportional to its volume.
10. Precautions
- Ensure that the temperature remains constant throughout the experiment.
- Check for any air leaks in the apparatus.
- Read the pressure gauge and volume scale accurately, avoiding parallax errors.
- Allow the system to stabilize after each pressure change before taking readings.
- Handle mercury with care as it is toxic. Use gloves if necessary.
- Ensure that the initial volume of air in the apparatus is sufficient for the experiment.
- Make sure the pressure gauge is calibrated correctly.
- Take at least 8-10 readings for accurate graphical analysis.
- Avoid rapid changes in pressure which might lead to temperature changes.
- Record all measurements promptly and accurately.
11. Sources of Error
- Temperature variations: Small changes in temperature during the experiment can affect the relationship between pressure and volume.
- Air leakage: Any leakage in the apparatus would result in incorrect volume readings.
- Parallax errors: Incorrect reading of volume or pressure scales due to viewing angle.
- Scale calibration errors: Inaccuracies in the pressure gauge or volume scale calibration.
- Mercury sticking to the walls: Can cause errors in volume readings.
- Non-ideal behavior of air: At high pressures or very low temperatures, real gases deviate from Boyle's Law.
- Instrument resolution limitations: The smallest division on the scales limits the precision of readings.
- Human reaction time: Delay in taking readings when the system is changing.
- Compression/expansion not being truly isothermal: Rapid compression/expansion can cause temperature changes.
12. Viva Voce Questions
Q1: State Boyle's Law. Who formulated it and when?
Boyle's Law states that at constant temperature, the pressure of a given mass of gas is inversely proportional to its volume. It was formulated by Robert Boyle in 1662 based on experiments conducted with air trapped in a J-shaped tube.
Q2: How is Boyle's Law expressed mathematically?
Mathematically, Boyle's Law is expressed as:
\[ P \propto \frac{1}{V} \] or \[ PV = \text{constant} \]
Where P is pressure and V is volume, at constant temperature and mass.
Q3: Why does the graph between P and V form a hyperbola?
The graph between P and V forms a hyperbola because pressure and volume are inversely proportional to each other (P ∝ 1/V). This inverse relationship mathematically produces a hyperbolic curve when plotted.
Q4: Why does the P vs 1/V graph give a straight line?
The P vs 1/V graph gives a straight line because according to Boyle's Law, P is directly proportional to 1/V (P ∝ 1/V or P = k × 1/V). This is the equation of a straight line passing through the origin with slope k.
Q5: How does the kinetic theory of gases explain Boyle's Law?
According to the kinetic theory of gases, when volume decreases at constant temperature, gas molecules are confined to a smaller space. This results in more frequent collisions of molecules with the container walls, increasing the pressure proportionally. Similarly, when volume increases, the frequency of collisions decreases, reducing pressure.
Q6: Under what conditions does a real gas deviate from Boyle's Law?
Real gases deviate from Boyle's Law under conditions of:
- Very high pressure, where molecules come close enough for intermolecular forces to become significant
- Very low temperature, where gas molecules are more likely to condense
- When the volume of the gas molecules themselves becomes significant compared to the container volume
Q7: How is Boyle's Law used in daily life or industrial applications?
Boyle's Law has many applications:
- Breathing and lung function (expansion/contraction causing pressure changes)
- Syringe operation
- Scuba diving equipment (regulator function)
- Tire pressure changes with volume
- Compressors and pumps
- Hydraulic systems
- Weather forecasting (pressure systems)
Q8: How is Boyle's Law related to the other gas laws?
Boyle's Law (PV = constant) relates pressure and volume at constant temperature and mass. It combines with:
- Charles's Law (V ∝ T at constant P): relates volume and temperature
- Gay-Lussac's Law (P ∝ T at constant V): relates pressure and temperature
- Avogadro's Law (V ∝ n at constant P, T): relates volume and moles
Q9: What would happen to the graph if there was a leak in the apparatus?
If there was a leak in the apparatus, the volume of gas would decrease unexpectedly as some gas escapes. This would cause deviations from the expected P-V relationship. The P-V product would not remain constant, and the graph of P vs 1/V would not be a straight line passing through the origin. The results would incorrectly suggest that Boyle's Law is not valid.
Q10: How would the results be affected if the temperature changed during the experiment?
If the temperature increased during the experiment, the gas would expand, increasing its volume for a given pressure. This would result in higher PV products than expected. If the temperature decreased, the gas would contract, resulting in lower PV products. Either way, the change in temperature would cause deviations from Boyle's Law, as the law is valid only at constant temperature. The P vs 1/V graph would no longer be a perfect straight line.