Mercury is poured into the open end of a J-shaped glass tube, which is closed at the short end, trapping air in that end. How much mercury can be poured in before the mercury overflows? Assume air to act like an ideal gas. The long and short arms are 1 m and 0.5 m long, respectively. Take atmospheric pressure to be 76 cm Hg.
Well, I hope the mercury doesn't have any stage fright! Let's see how much it can pour before the overflow.
To find the amount of mercury that can be poured, we need to consider the pressure difference between the trapped air and the atmosphere.
In the long arm of the J-shaped tube, the pressure exerted by the column of mercury is equal to the atmospheric pressure (76 cm Hg) plus the pressure exerted by the trapped air. We can use the ideal gas law to find the pressure exerted by the trapped air.
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
Since the glass is closed at the short end, the volume of the trapped air remains constant. Therefore, the pressure exerted by the trapped air is directly proportional to the number of moles of gas present.
The pressure in the short arm of the tube is determined by the height of the mercury column in the inverted arm. Since the height of the column is 0.5 m, we can convert it to pressure using the density of mercury (13.6 g/cm3).
So, the pressure exerted by the trapped air is equal to the atmospheric pressure (76 cm Hg) plus the pressure of the mercury column in the short arm.
Now, let's calculate the pressure exerted by the trapped air:
Pressure = atmospheric pressure + pressure due to height of mercury column
= 76 cm Hg + (0.5 m * density of mercury * acceleration due to gravity)
≈ 76 cm Hg + (0.5 m * 13.6 g/cm3 * 9.8 m/s2)
To determine the amount of mercury that can be poured, we compare the pressure exerted by the trapped air to the atmospheric pressure (76 cm Hg). If the pressure exerted by the trapped air is greater than the atmospheric pressure, the mercury will overflow.
So, if the pressure exerted by the trapped air is less than or equal to the atmospheric pressure, the mercury won't overflow. Otherwise, it will overflow.
To determine how much mercury can be poured into the J-shaped glass tube before it overflows, we need to consider the pressure difference between the long and short arms of the tube.
1. First, we need to convert the atmospheric pressure from cm Hg to meters of mercury (Hg).
- 1 cm Hg = 0.01 m Hg
- Therefore, atmospheric pressure = 76 cm Hg * 0.01 m Hg/cm Hg = 0.76 m Hg
2. Next, we can calculate the pressure at the bottom of the long arm of the J-shaped tube.
- The pressure at the bottom is equal to the atmospheric pressure plus the pressure due to the height of the column of mercury.
- Pressure = atmospheric pressure + (density of mercury * gravitational acceleration * height)
Density of mercury = 13,600 kg/m³
Gravitational acceleration = 9.8 m/s²
Height of the long arm = 1 m
Pressure = 0.76 m Hg + (13,600 kg/m³ * 9.8 m/s² * 1 m) = 0.76 m Hg + 133,280 Pa
3. Now, let's calculate the pressure at the bottom of the short arm of the J-shaped tube.
- The pressure at the bottom is equal to the atmospheric pressure.
Pressure = atmospheric pressure = 0.76 m Hg
4. Since we have the pressure at the bottom of both arms, we can compare them.
- If the pressure at the bottom of the long arm is greater than the pressure at the bottom of the short arm, the mercury will not overflow.
- If the pressure at the bottom of the long arm is less than the pressure at the bottom of the short arm, the mercury will overflow.
In this case, the pressure at the bottom of the long arm is (0.76 m Hg + 133,280 Pa), and the pressure at the bottom of the short arm is 0.76 m Hg.
So, if (0.76 m Hg + 133,280 Pa) > 0.76 m Hg, the mercury will overflow.
Therefore, the maximum height of mercury that can be poured into the open end of the J-shaped tube without overflowing depends on the pressure difference between the two arms of the tube and cannot be determined without additional information.
To calculate how much mercury can be poured into the J-shaped glass tube before it overflows, we need to consider the difference in pressure between the long and short arms of the tube.
The pressure difference can be determined using the equation:
ΔP = ρgh
Where:
ΔP is the pressure difference
ρ is the density of mercury
g is the acceleration due to gravity
h is the height difference between the two arms of the tube
In this case, the long arm of the tube is 1 m long, while the short arm is 0.5 m long. We can use these values to calculate the height difference (h) as follows:
h = 1 m - 0.5 m
h = 0.5 m
Next, we need to determine the density of mercury. The density of mercury is approximately 13,595 kg/m³.
Using these values, we can now calculate the pressure difference:
ΔP = ρgh
ΔP = (13,595 kg/m³) * (9.8 m/s²) * (0.5 m)
ΔP = 66,513.1 N/m²
Now, we need to convert the pressure to centimeters of mercury (cm Hg). We can use the conversion factor that 1 cm Hg is equivalent to 1333.22 N/m².
Pressure in cm Hg = ΔP / (1333.22 N/m² per cm Hg)
Pressure in cm Hg = 66,513.1 N/m² / 1333.22 N/m² per cm Hg
Pressure in cm Hg = 49.82 cm Hg
Finally, to determine the maximum height of mercury that can be poured into the tube without overflowing, we need to subtract the atmospheric pressure from the calculated pressure difference:
Maximum height of mercury = Pressure difference - Atmospheric pressure
Maximum height of mercury = 49.82 cm Hg - 76 cm Hg
Maximum height of mercury = -26.18 cm Hg
Since the result is negative, it means that the atmospheric pressure is greater than the pressure difference. Therefore, no mercury can be poured into the tube without overflowing.