1. A student collected the following data for a fixed volume of gas:
Temperature (⁰C) Pressure (mm of Hg)
10 726
20 750
40 800
70 880
100 960
150 ???
Fill in the missing data point. Show all calculations leading to an answer.
(750-726)/(20-10) = 2.4
(800-750)/(40-20) = 2.5
(880-800)/(70-40) = 2.7
(960-880)/(100-70)= 2.7
slope increasing a bit but 2.7 seems a reasonable guess
slope = (x-960)/(150-100)
(x-960)= 50(2.7)
x = 960 + 135 = 1095
To fill in the missing data point in a table of temperature and pressure for a fixed volume of gas, we can use the concept of Charles's Law. Charles's Law states that, for a fixed amount of gas at constant pressure, the volume of the gas is directly proportional to its temperature (assuming pressure remains constant).
To determine the missing data point (pressure at 150⁰C), we can use the known values and create a linear equation to solve for the missing value. Here's how to do it:
1. First, let's plot the known data points on a graph. Place the temperature values on the x-axis and the pressure values on the y-axis. Connect the points to create a straight line.
2. Next, determine the equation of the line. To do this, find the equation of a straight line that fits the data points. The equation of a straight line can be written as y = mx + b, where y is the dependent variable (pressure), x is the independent variable (temperature), m is the slope, and b is the y-intercept.
3. Find the slope (m). The slope of the line can be found using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two coordinates on the line. Select two points from the known data points and calculate the slope.
Using the points (10, 726) and (40, 800), we can calculate the slope:
m = (800 - 726) / (40 - 10)
m = 74 / 30
m = 2.47
4. With the slope found, we can determine the y-intercept (b) using the equation of a straight line. Use one known data point and the slope to solve for b.
Using the point (10, 726):
726 = 2.47 * 10 + b
726 = 24.7 + b
b = 726 - 24.7
b = 701.3
5. Now that we have the equation of the line, we can use it to find the missing data point. Substitute the missing temperature value into the equation and solve for pressure:
Let the missing temperature be 150⁰C:
Pressure = 2.47 * 150 + 701.3
Pressure = 370.5 + 701.3
Pressure = 1,071.8
Therefore, the estimated pressure at 150⁰C is 1,071.8 mm of Hg.