According to Charles’ Law, the pressure P in pascals of a fixed volume of a gas is linearly related to the temperature T in degrees Celsius. In an experiment, it was found that when T = 40, P = 90 and when T = 80, P = 100.
a. What is the slope of the line containing these points?
b. Explain the meaning of the slope in this context.
c. Graph the function
The correct answer and explanation is :
Let’s work through this problem step-by-step.
Part (a): Finding the slope of the line
The formula for the slope (m) of a line between two points ((T_1, P_1)) and ((T_2, P_2)) is given by:
[
m = \frac{P_2 – P_1}{T_2 – T_1}
]
From the given data, we know:
- When (T = 40), (P = 90) (point 1: (T_1 = 40), (P_1 = 90))
- When (T = 80), (P = 100) (point 2: (T_2 = 80), (P_2 = 100))
Now, substitute these values into the slope formula:
[
m = \frac{100 – 90}{80 – 40} = \frac{10}{40} = 0.25
]
So, the slope of the line is 0.25.
Part (b): Meaning of the slope
The slope (0.25) in this context represents the rate of change of pressure (P) with respect to temperature (T). In other words, for every increase of 1 degree Celsius in temperature, the pressure increases by 0.25 pascals. This slope is a direct indication of how sensitive the pressure is to changes in temperature for the fixed volume of gas, according to Charles’ Law.
Part (c): Graphing the function
The linear relationship between pressure and temperature can be written as:
[
P = mT + b
]
where (m = 0.25) is the slope, and (b) is the y-intercept. To find the value of (b), we can use one of the points. Let’s use the point ((T_1 = 40, P_1 = 90)) and substitute into the equation:
[
90 = 0.25(40) + b
]
Simplifying:
[
90 = 10 + b
]
[
b = 90 – 10 = 80
]
Thus, the equation of the line is:
[
P = 0.25T + 80
]
This is the equation that relates pressure and temperature for this gas.
Graph of the function
To graph this equation, we will plot the points ((40, 90)) and ((80, 100)) on a coordinate plane and draw a straight line through them. The line will have a slope of (0.25) and will intersect the pressure axis (P-axis) at (P = 80).
Let’s summarize the key points:
- Slope: (0.25) means for each 1°C increase in temperature, pressure increases by 0.25 pascals.
- Equation: The relationship between pressure and temperature is linear and can be described by (P = 0.25T + 80).
- Graph: The line will pass through the points ((40, 90)) and ((80, 100)), with a slope of 0.25 and y-intercept at (P = 80).