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Chapter 5: Problem 21

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

Expert verified
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

01

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.
02

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 \[ 101325 = 13600 \cdot 9.81 \cdot h_m. \]
03

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: \[ h_m = \frac{101325}{13600 \cdot 9.81} = 0.760m. \]
04

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.
05

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.

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