Question

If a barometer were built using water \(\left(d=1.0 \mathrm{~g} / \mathrm{cm}^{3}\right)\) instead of mercury \(\left(d=13.6 \mathrm{~g} / \mathrm{cm}^{3}\right)\), would the column of water be higher than, lower than, or the same as the column of mercury at \(1.00\) atm? If the level is different, by what factor? Explain.

Short answer

If a barometer were built using water instead of mercury, the column of water would be higher than the column of mercury at 1 atm. The water column would be approximately 13.6 times higher than the mercury column, because water has a lower density than mercury, and a greater height of water is needed to exert the same pressure as a shorter column of the denser mercury.

Step-by-step solution

Recall the formula for pressure at a certain depth in a fluid

The formula we will use is: \[ P = ρgh, \] where "P" is the pressure, "ρ" is the fluid density, "g" is the gravitational acceleration (approximately 9.81 m/s²), and "h" is the fluid column height.

Set up the equation for water and mercury at 1 atm

At 1 atm (101325 Pa), we want to find the heights of both water and mercury columns. We need to convert the densities given in the problem to SI units to be able to plug into the formula. The density of water is given as 1.0 g/cm³, which is equal to \( 1000 \frac{kg}{m^3} \) after conversion. The density of mercury is given as 13.6 g/cm³, which is equal to \( 13600 \frac{kg}{m^3} \) after conversion. So we can find the height of water and mercury columns by setting up the following equations: For water: \[ P_w = ρ_wgh_w, \] or \[ 101325= 1000 \cdot 9.81 \cdot h_w. \] For mercury: \[ P_m = ρ_mgh_m, \] or \[ 101325 = 13600 \cdot 9.81 \cdot h_m. \]

Solve for the heights of water and mercury columns at 1 atm

First, we'll solve for the height of the water column: \[ h_w = \frac{101325}{1000 \cdot 9.81} = 10.33m. \] Next, we'll solve for the height of the mercury column: \[ h_m = \frac{101325}{13600 \cdot 9.81} = 0.760m. \]

Compare the heights and find the factor of difference

Comparing the heights, we see that the water column (10.33 m) is higher than the mercury column (0.760 m). To find the factor by which they differ, we can divide the height of the water column by the height of the mercury column: \[ \text{Factor} = \frac{h_w}{h_m} = \frac{10.33}{0.760} ≈ 13.6. \] So, the water column is 13.6 times higher than the mercury column at 1 atm.

Conclusion

In conclusion, if a barometer were built using water instead of mercury, the column of water would be higher than the column of mercury at 1 atm. The water column would be approximately 13.6 times higher than the mercury column. This is because the density of water is lower than the density of mercury, so a greater height of water is needed to exert the same pressure as a shorter column of the denser mercury.

Key concepts

Understanding Barometers

A barometer is an instrument that measures atmospheric pressure. It uses fluid columns and reflects changes in pressure by the height of the fluid column.

In a typical barometer, mercury is used because of its high density. Mercury barometers consist of a glass tube filled with mercury, inverted over a mercury reservoir.

If atmospheric pressure increases, it pushes more mercury into the tube, raising the mercury column's height. Conversely, if the pressure decreases, the column height lowers. Essentially, this change in mercury height directly correlates with atmospheric pressure changes.

Now, if we consider using water instead of mercury in a barometer, the fluid dynamics change due to water's lower density. Water barometers require taller columns to balance the atmospheric pressure, as seen in the given exercise. This height difference is a direct consequence of water being much less dense than mercury, producing 13.6 times the height as calculated in the problem's solution.

Influence of Fluid Density

Fluid density is a fundamental factor affecting fluid pressure in a barometer. Density is the mass per unit volume of a substance.

In the context of our exercise, fluid density ( \(\rho\)) is critical because it determines how much of a fluid column is needed to exert a certain level of pressure at its base. Lower density means more fluid is needed.

• Water, with a density of \(1000 \, \text{kg/m}^3\), is less dense than mercury, which has a density of \(13600 \, \text{kg/m}^3\).
• This means a column of water must be significantly taller than a mercury column to exert the same pressure at its base.

Calculating fluid pressure ( \(P = \rho gh\)), shows water’s larger column size due to its lower density.

Therefore, fluid density dictates the height of the liquid column in a barometer when exposed to a constant pressure, as lower-density fluids require taller columns to balance the same atmospheric pressure.

Role of Gravitational Acceleration

Gravitational acceleration ( \(g\)) is a constant value that plays a critical role in calculating fluid pressure. On Earth, this constant is approximately \(9.81 \, \text{m/s}^2\).

When calculating pressure in fluid dynamics, gravitational acceleration helps determine the pressure exerted by a fluid column. The formula used in our context is \(P = \rho gh\), where each variable contributes to the overall pressure calculation.

• \(\rho\) is the density of the fluid.
• \(h\) is the height of the fluid column.
• \(g\) remains constant as it represents the force exerted by Earth's gravity on the fluid.

Because \(g\) is constant, changes in either fluid density or column height are responsible for variations in pressure measurements in barometers.

Understanding how gravitational acceleration interacts with fluid density and column height is crucial for explaining why, under constant atmospheric conditions, denser fluids necessitate less height to exert the same pressure as a less dense fluid.