How high in meters must a column of glycerol be to exert a pressure equal to that of a 760-mm column of mercury? The density of glycerol is 1.26 g/mL, whereas that of mercury is 13.6 g/mL.

Verified step by step guidance

1. Identify the relationship between pressure, density, and height for a liquid column: \( P = \rho g h \), where \( P \) is pressure, \( \rho \) is density, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column.
2. Recognize that the pressure exerted by the glycerol column must equal the pressure exerted by the mercury column. Therefore, set the pressures equal: \( \rho_{\text{glycerol}} g h_{\text{glycerol}} = \rho_{\text{mercury}} g h_{\text{mercury}} \).
3. Cancel out the gravitational acceleration \( g \) from both sides of the equation, as it is a common factor: \( \rho_{\text{glycerol}} h_{\text{glycerol}} = \rho_{\text{mercury}} h_{\text{mercury}} \).
4. Substitute the given densities and the height of the mercury column into the equation: \( 1.26 \text{ g/mL} \times h_{\text{glycerol}} = 13.6 \text{ g/mL} \times 760 \text{ mm} \).
5. Solve for \( h_{\text{glycerol}} \) by dividing both sides by the density of glycerol: \( h_{\text{glycerol}} = \frac{13.6 \times 760}{1.26} \). Convert the result from millimeters to meters by dividing by 1000.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. Understanding this concept is essential for comparing the pressures exerted by different fluids at varying heights.

Density

Density is defined as mass per unit volume and is a critical property of substances that influences how they behave under pressure. In this context, the densities of glycerol and mercury are necessary to determine how high a column of glycerol must be to exert the same pressure as a column of mercury. The density of glycerol is 1.26 g/mL, while mercury has a density of 13.6 g/mL, indicating that mercury is much denser.

Pressure Conversion

Pressure conversion involves translating pressure measurements from one unit to another, such as from mm of mercury (mmHg) to pascals (Pa) or other units. In this problem, the pressure exerted by a 760-mm column of mercury must be understood in terms of the equivalent height of glycerol. This requires knowledge of the relationship between the heights of fluid columns and their respective densities.