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Chapter 3: Problem 34

A barometer accidentally contains 6.5 inches of water on top of the mercury column (so there is also water vapor instead of a vacuum at the top of the barometer). On a day when the temperature is \(70^{\circ} \mathrm{F}\), the mercury column height is 28.35 inches (corrected for thermal expansion). Determine the barometric pressure in psia. If the ambient temperature increased to \(85^{\circ} \mathrm{F}\) and the barometric pressure did not change, would the mercury column be longer, be shorter, or remain the same length? Justify your answer.

Short Answer

Expert verified
The barometric pressure, at \(70^{\circ} \mathrm{F}\), is determined by calculating the pressures exerted by the mercury, water, and vapor and adding these together. If the temperature were to increase to \(85^{\circ} \mathrm{F}\) without a change in barometric pressure, the mercury column would have to increase in length due to decreases in the densities of the mercury and water.

Step by step solution

01

Calculate Barometric Pressure at \(70^{\circ} \mathrm{F}\)

We first need to calculate the barometric pressure at \(70^{\circ} \mathrm{F}\), which consists of adding three parts: the pressure exerted by the mercury, water, and water vapor. Use the equation \(P = P_{\text{mercury}} + P_{\text{water}} + P_{\text{vapor}}\), where \(P_{\text{mercury}} = h_{\text{mercury}} \cdot \rho_{\text{mercury}} \cdot g\), \(P_{\text{water}} = h_{\text{water}} \cdot \rho_{\text{water}} \cdot g\), and \(P_{\text{vapor}}\) is calculated from steam tables. Use the given heights for the mercury and water (28.35 inches and 6.5 inches, respectively), the gravity constant g, and the densities of mercury and water at 70 \(^\circ \mathrm{F}\).
02

Predict Changes to Mercury Column Length at \(85^{\circ} \mathrm{F}\)

Assuming the barometric pressure does not change, we need to consider how a temperature increase to \(85^{\circ} \mathrm{F}\) would affect the mercury column length. Since the density of the mercury and water would both decrease with an increase in temperature (based on the ideal gas law), the height of the mercury column would have to increase in order for the pressures exerted by the liquid column and the surrounding air pressure to re-balance. Therefore, the mercury column would be longer at the increased temperature.

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