Boyle's and Charles's Gas Law Applications

Boyle's Law describes the relationship between the pressure and volume of a gas.
In this case, the temperature ( $T$ ) and the number of moles ( $n$ ) are held constant.
The relationship between pressure and volume is an inverse relationship. This means that as one variable increases, the other decreases.
- For example, if the pressure increases, the volume must decrease.
- Conversely, if the pressure decreases, the volume must increase.
A general rule of thumb described is the "one to one relationship": if the pressure is doubled, the volume is generally halved. However, the speaker notes that it "doesn't always work that way," but the general inverse trend remains the same.
The formula for Boyle's Law is expressed as: $P_1 V_1 = P_2 V_2$

Scenario Parameters:
- Initial Volume ( $V_1$ ): $1.53\,\text{L}$ of $SO_2$ .
- Initial Pressure ( $P_1$ ): $5.6 \times 10^3$ pistons (noted later in calculation as the same units as $P_2$ ).
- Final Pressure ( $P_2$ ): Increases to $1.5 \times 10^4\,\text{pascals}$ .
- Condition: Constant temperature.
Solving for New Volume ( $V_2$ ):
- To find the new volume, the equation is rearranged to solve for the unknown variable: $V_2 = \frac{P_1 V_1}{P_2}$
- Substituting the values: $V_2 = \frac{(5.6 \times 10^3) \times (1.53)}{1.5 \times 10^4}$
Units and Significant Figures:
- The units for pressure cancel out if they are the same.
- The initial volume is in liters, so the final answer will be in liters.
- Based on the precision of the input data, the final answer must be rounded to two significant figures.
- Calculation result: $0.57\,\text{L}$ .
Verification: The pressure increased from $5.6 \times 10^3$ to $1.5 \times 10^4$ , so the volume decreased from $1.53\,\text{L}$ to $0.57\,\text{L}$ , confirming the inverse relationship.

Charles's Law defines the derivation where the ratio of volume to temperature is constant: $\frac{V}{T} = k$
This results in the formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Linear Form Preference: The speaker prefers expressing the equation in linear form to make substituting and solving for variables easier: $V_1 \times T_2 = V_2 \times T_1$
Constants: For Charles's Law to apply, the pressure ( $P$ ) and the number of moles ( $n$ ) must remain constant. If they change, this law cannot be used.
Direct Relationship: Charles's Law is a direct relationship. If one variable increases, the other increases as well.
- Physical Explanation: If a container of gas is heated, the particles start moving more rapidly. These moving particles push outward, necessitating more space and causing the volume to increase.
Critical Temperature Rule: All temperatures used in gas law calculations must be converted to Kelvin by adding $273$ to the Celsius value.

Example 1: Initial Volume and Temperature Change

Data:
- Initial Temperature ( $T_1$ ): $25^\circ\text{C}$ . In Kelvin: $25 + 273 = 298\,\text{K}$ .
- Final Temperature ( $T_2$ ): $95^\circ\text{C}$ . In Kelvin: $95 + 273 = 368\,\text{K}$ .
Solving for $V_2$ :
- Multiply both sides of the standard equation by $T_2$ : $V_2 = \frac{V_1 \times T_2}{T_1}$
Significant Figures Note:
- The raw calculated answer is $308.7$ .
- Because volume one ( $V_1$ ) only has two significant figures, the final answer must be adjusted for precision.
- Final Answer: $3.0 \times 10^2$ .
- The speaker acknowledges that while the precision rules might make it seem like the number is changing significantly, this is the best that can be done based on the given precision.

Example 2: Volume at $38^\circ\text{C}$

Data:
- Gas at $15^\circ\text{C}$ with a volume of $2.58\,\text{L}$ .
- Initial Temperature ( $T_1$ ): $15 + 273 = 288\,\text{K}$ .
- Initial Volume ( $V_1$ ): $2.58\,\text{L}$ .
- Final Temperature ( $T_2$ ): $38^\circ\text{C}$ . In Kelvin: $38 + 273 = 311\,\text{K}$ .
Calculation:
- Goal: Find $V_2$ .
- Formula: $V_2 = \frac{V_1 \times T_2}{T_1}$ $V_2 = \frac{2.58 \times 311}{288}$
- Final Answer: $2.79\,\text{L}$ .
Verification: Since the temperature went up, the volume also went up ( $2.58\,\text{L}$ to $2.79\,\text{L}$ ), proving the direct relationship.

Inverse vs. Direct Plots:
- Boyle's Law: When plotting an inverse relationship (Pressure vs. Volume), the result is a curve.
- Charles's Law: When plotting a direct relationship (Volume vs. Temperature), the result is a straight line.
Absolute Temperature:
- If you plot Volume versus Temperature for different amounts of gases, all lines converge at a specific point where the volume equals zero.
- This point is defined as "absolute temperature."
- The speaker identifies this point as "zero degrees Celsius or minus two seventy three point five Kelvins."
- This temperature is described as very cold and extremely difficult to achieve.