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.

In a fluid at rest, the pressure at any given depth depends on fluid density and the height of the column above that point. The hydrostatic pressure is given by the formula: \[P = hρg\], where \(P\) is the hydrostatic pressure, \(h\) is the height of the fluid column, \(ρ\) is the fluid density, and \(g\) is the acceleration due to gravity.

Using the hydrostatic pressure formula for mercury, we can find the height of the column by replacing pressure with the atmospheric pressure (1.00 atm), and using mercury's density and gravity's acceleration: \[ h_{Hg} = \frac{P_{atm}}{ρ_{Hg} \times g} \] Given that the density of mercury is 13.6 g/cm³, and 1 atm = 101,325 Pa (Pascals) and g = 9.81 m/s², we can now calculate the height: \[ h_{Hg} = \frac{101,325}{13.6 \times 10^{-3} \times 9.81} \] \[ h_{Hg} \approx 76.0 \mathrm{cm} \]

Similarly, we will calculate the height of the water column using the same formula: \[ h_{H_2O} = \frac{P_{atm}}{ρ_{H_2O} \times g} \] Given that the density of water is 1.0 g/cm³, we can now calculate the height in a similar fashion as the mercury column: \[ h_{H_2O} = \frac{101,325}{1.0 \times 10^{-3} \times 9.81} \] \[ h_{H_2O} \approx 10,336.0 \mathrm{cm} \]

Now that we have the heights of both mercury and water columns, we can compare them: \[ h_{H_2O} \approx 10,336.0 \mathrm{cm} \] (Water column) \[ h_{Hg} \approx 76.0 \mathrm{cm} \] (Mercury column) The water column is higher by a considerable amount. To find the factor by which they differ, divide the height of the water column by the height of the mercury column: \[ factor = \frac{h_{H_2O}}{h_{Hg}} \] \[ factor \approx \frac{10,336.0}{76.0} \] \[ factor \approx 136 \] So, at 1.00 atm, the column of water is higher than the mercury column by a factor of approximately 136. This difference is due to the significantly lower density of water compared to mercury, resulting in a much taller column height needed to balance the atmospheric pressure.