The density of liquid gallium is 6.09 g/cm3 at 35°C. If this element is employed in a barometer instead of mercury, what is the height of a column of gallium sustained in the barometer at 1 atm pressure?
The Correct Answer and Explanation is :
To solve this problem, we will use the concept that the pressure exerted by a column of liquid in a barometer is given by the equation: P=ρ⋅g⋅hP = \rho \cdot g \cdot h
Where:
- PP is the pressure exerted by the liquid column (1 atm = 101325 Pa),
- ρ\rho is the density of the liquid (in kg/m³),
- gg is the acceleration due to gravity (9.81 m/s²),
- hh is the height of the liquid column (in meters).
Step 1: Convert Density of Gallium to SI Units
The density of liquid gallium is given as 6.09 g/cm³ at 35°C. We need to convert this density into kg/m³ for use in the equation.
1 g/cm³ = 1000 kg/m³
Thus, the density of gallium: ρ=6.09 g/cm3=6.09×1000 kg/m3=6090 kg/m3\rho = 6.09 \, \text{g/cm}^3 = 6.09 \times 1000 \, \text{kg/m}^3 = 6090 \, \text{kg/m}^3
Step 2: Rearranging the Equation for Height
We can rearrange the pressure equation to solve for the height hh: h=Pρ⋅gh = \frac{P}{\rho \cdot g}
Substituting the known values: h=101325 Pa6090 kg/m3⋅9.81 m/s2h = \frac{101325 \, \text{Pa}}{6090 \, \text{kg/m}^3 \cdot 9.81 \, \text{m/s}^2}
Step 3: Calculation
h=1013256090⋅9.81≈10132559758.9≈1.695 mh = \frac{101325}{6090 \cdot 9.81} \approx \frac{101325}{59758.9} \approx 1.695 \, \text{m}
Thus, the height of the column of gallium sustained in the barometer at 1 atm pressure is approximately 1.695 meters.
Explanation
In a barometer, the height of the liquid column corresponds to the atmospheric pressure. Since mercury, which is commonly used in barometers, has a much higher density than gallium, the column of gallium will be taller for the same pressure. The density of gallium being much lower than mercury (which has a density of about 13,600 kg/m³) results in a taller column to balance the same atmospheric pressure. Therefore, replacing mercury with gallium in a barometer results in a significant increase in the height of the liquid column.