A uniform capillary tube closed at one end contained dry air trapped by a thread of mercury 8.5×10^-2m long.When the tube was held horizontally the length of the air column was 5×10^-2m,when it was held vertically with the closed end downwards the length was 4.5×10^-2m.Determine the value of the atmospheric pressure.g=10m/s, density of mercury 1.36×10^4kg/m^-3
Do it again
find the weight of the column of Hg (density*height*area).
pressure then from that pressure at base of column=density*height+1atm
so, the air trapped underneight it has the same pressure. Find the moles in the column vertically
n=PV/RT where v=length(area). Leave Area as a cunit, it will divide away soon.
now horizontally, same number of moles, but Pressure is now just 1 atm. What is the volume: Vnew=lengthnew*area
V=Length*area=nRT/1atm= you do it, solve for length
To determine the value of atmospheric pressure, we need to consider the pressure difference between the two situations: when the tube is held horizontally and when it is held vertically.
Step 1: Calculate the pressure at the closed end of the capillary tube when held horizontally.
We can use the hydrostatic pressure formula: P = P₀ + ρgh, where P is the pressure, P₀ is the atmospheric pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.
Substituting the given values into the formula:
P₁ = P₀ + ρ₁gh₁
P₁ = P₀ + (1.36×10^4 kg/m³) × (10 m/s²) × (8.5 × 10⁻² m)
Step 2: Calculate the pressure at the closed end of the capillary tube when held vertically.
Similarly, using the hydrostatic pressure formula:
P₂ = P₀ + ρ₂gh₂
P₂ = P₀ + (1.36×10^4 kg/m³) × (10 m/s²) × (4.5 × 10⁻² m)
Step 3: Determine the atmospheric pressure.
Since the air is trapped inside the capillary tube, the pressure at the closed end in both situations is the same, i.e., P₁ = P₂.
Equating the two pressure equations:
P₀ + (1.36×10^4 kg/m³) × (10 m/s²) × (8.5 × 10⁻² m) = P₀ + (1.36×10^4 kg/m³) × (10 m/s²) × (4.5 × 10⁻² m)
Step 4: Solve for P₀.
By canceling out the terms and rearranging the equation:
P₀ = (1.36×10^4 kg/m³) × (10 m/s²) × (8.5 × 10⁻² m) - (1.36×10^4 kg/m³) × (10 m/s²) × (4.5 × 10⁻² m)
Step 5: Calculate the value of P₀.
Using a calculator, evaluate the expression above:
P₀ = (1.36×10^4 kg/m³) × (10 m/s²) × (8.5 × 10⁻² m) - (1.36×10^4 kg/m³) × (10 m/s²) × (4.5 × 10⁻² m)
P₀ ≈ 1704 Pa
Therefore, the value of the atmospheric pressure is approximately 1704 Pa.
To determine the value of the atmospheric pressure, we can use the equation of hydrostatic pressure:
P = P₀ + ρgh
Where:
- P is the pressure at a certain point
- P₀ is the pressure at a reference point (in this case, the pressure at the top of the column of air when held horizontally)
- ρ is the density of the fluid (in this case, the density of mercury)
- g is the acceleration due to gravity
- h is the height difference between the two points
Using the given values:
- P₀: The pressure at the top of the column of air when held horizontally can be considered as atmospheric pressure.
- ρ: The density of mercury is given as 1.36×10^4 kg/m^3.
- g: The acceleration due to gravity is given as 10 m/s^2.
- h: The height difference between the two points is (8.5×10^-2 - 5×10^-2)m when held horizontally and (8.5×10^-2 - 4.5×10^-2)m when held vertically.
Let's calculate the atmospheric pressure using the given values:
For the horizontal position:
P = P₀ + ρgh
P = P₀ + (1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 5×10^-2)m
For the vertical position:
P = P₀ + ρgh
P = P₀ + (1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 4.5×10^-2)m
By subtracting the equation for the vertical position from the equation for the horizontal position, we can eliminate the atmospheric pressure (P₀) and solve for P:
(P - P) = (P₀ - P₀) + [(1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 5×10^-2)m] - [(1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 4.5×10^-2)m]
Simplifying the equation, we get:
0 = [(1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 5×10^-2)m] - [(1.36×10^4 kg/m^3) × (10 m/s^2) × (8.5×10^-2 - 4.5×10^-2)m]
Now, we can solve for P:
(P₁ - P₂) = [(1.36×10^4 kg/m^3) × (10 m/s^2)] × [(8.5×10^-2 - 5×10^-2)m - (8.5×10^-2 - 4.5×10^-2)m]
Simplifying the equation further:
0 = (1.36×10^4 kg/m^3) × (10 m/s^2) × [(8.5×10^-2 - 5×10^-2)m - (8.5×10^-2 - 4.5×10^-2)m]
By calculating the right-hand side of the equation, we can determine the value of the atmospheric pressure.