A barometer has air trapped above the mercury in an inverted tube with the opening in a reservoir of mercury.
When the length l of air trapped is 10.0cm, the mercury height h is 72.0. When the tube is slightly lowered into the reservoir, l becomes 8.0cm. What is the new h?
Take atmospheric pressure to be equivalent to 76.0cm mercury.
To find the new height of the mercury, we can use the concept of atmospheric pressure and relate it to the heights of the air and mercury columns in the barometer.
First, let's start by understanding the initial conditions given:
- Length of air trapped above the mercury column, l = 10.0 cm
- Height of the mercury column, h = 72.0 cm
- Atmospheric pressure = 76.0 cm mercury
In a barometer, the atmospheric pressure acts on the mercury column, pushing it up in the tube until the pressure of the trapped air equals the atmospheric pressure. This creates equilibrium.
When the tube is lowered, the length of the trapped air column decreases from 10.0 cm to 8.0 cm. We need to determine the new height of the mercury column.
To solve this, we'll use the principle that the atmospheric pressure acting on the column is equal to the pressure exerted by the trapped air plus the pressure exerted by the mercury column.
The pressure exerted by the trapped air can be calculated using Boyle's Law, which states that the pressure and volume of a gas are inversely proportional, assuming constant temperature:
P1 * V1 = P2 * V2
Where:
P1 = Initial pressure (atmospheric pressure)
P2 = Final pressure (pressure exerted by the trapped air)
V1 = Initial volume (length of the air column initially trapped, l)
V2 = Final volume (length of the air column after the tube is lowered, l')
Since the temperature is constant, we can rewrite the equation as:
P1 * l = P2 * l'
We can solve this equation for P2:
P2 = (P1 * l) / l'
Now, we can determine the pressure exerted by the mercury column using the hydrostatic pressure formula:
P_mercury = Density_mercury * g * h
Where:
Density_mercury = Density of mercury
g = Acceleration due to gravity
h = Height of the mercury column
The density of the mercury is given, so we can calculate the pressure exerted by the mercury column using the given value of h.
Finally, the new height of the mercury column, h', can be determined by finding the height that corresponds to the pressure P2, which is the sum of the pressure exerted by the trapped air (P2) and the pressure exerted by the mercury column (P_mercury):
P2 + P_mercury = P_atmospheric
Let's calculate the new height of the mercury column, h':
Step 1: Calculate P2
P2 = (P1 * l) / l'
Step 2: Calculate P_mercury
P_mercury = Density_mercury * g * h
(where Density_mercury = density of mercury = 13.6 g/cm³, g = acceleration due to gravity = 9.8 m/s²)
Step 3: Calculate P_atmospheric
P_atmospheric = atmospheric pressure = 76.0 cm mercury
Step 4: Calculate the new height, h'
h' = (P_atmospheric - P2) / P_mercury
By substituting the calculated values into the above equation, we can find the new height of the mercury column, h'.
To find the new height of the mercury, we can use the Boyle's Law, which states that the product of pressure and volume is constant at constant temperature.
Let's assume the initial pressure inside the trapped air column is equal to the atmospheric pressure (76.0 cm of mercury).
Using Boyle's Law, we can set up the following equation:
P1 x V1 = P2 x V2
Where:
P1 = initial pressure (atmospheric pressure) = 76.0 cm of mercury
V1 = initial volume of air trapped = length of air trapped (l1) = 10.0 cm
P2 = final pressure (pressure due to the height of the mercury) = ?
V2 = final volume of air trapped = length of air trapped (l2) = 8.0 cm
Substituting the given values into the equation:
76.0 cm x 10.0 cm = P2 x 8.0 cm
Simplifying the equation:
760 cm^2 = 8P2 cm^2
Next, we solve for P2:
P2 = 760 cm^2 / 8 cm^2
P2 = 95.0 cm of mercury
Therefore, the new height of the mercury (h2) is equal to the pressure due to the length of the trapped air column:
h2 = P2 - atmospheric pressure
h2 = 95.0 cm - 76.0 cm
h2 = 19.0 cm
So, the new height of the mercury is 19.0 cm.