a. Why does a column of mercury in a tube that is inverted in a dish of mercury have a height of about 760 mm at sea level? b. The density of water is approximately 1/13.5 the density of mercury. What height would be maintained by a column of water inverted in a dish of water at sea level? c. What accounts for the difference in the heights of the mercury and water columns?

Atmospheric pressure is the force exerted by the weight of the atmosphere above us. Imagine the atmosphere as a giant blanket of air covering the earth. This blanket pushes down on everything at the surface. At sea level, this pressure is about 101,325 Pascals (Pa). This value means that over each square meter, the atmosphere exerts a force of 101,325 Newtons.

When we talk about the height of a mercury column, we're discussing how much force is required to balance the weight of the air column above us. The mercury column height is 760 mm because this precisely balances the atmospheric pressure exerted at sea level. Without this balance, the mercury would either rise higher or fall, depending on the external pressure conditions.

Fluid density refers to how much mass a fluid has in a specific volume. It's like asking how tightly packed the molecules in the fluid are. Mercury, for example, has a very high density, meaning it’s very heavy for its size. Water has a lower density, meaning it’s lighter compared to mercury for the same volume.

The density of water is approximately 1/13.5th of the density of mercury. This big difference means we need more water to exert the same pressure as a smaller amount of mercury. When measuring how high water or mercury rises in a column due to pressure, density plays a crucial role. Less dense fluids (like water) need to rise higher to balance a given pressure compared to denser fluids (like mercury). Hence, while a mercury column can balance atmospheric pressure at 760 mm, a water column would need to be about 10.26 meters high!

The pressure equation is key in understanding how different fluids behave under pressure. The equation is expressed as: \[ P = \rho gh \] where:

• P stands for pressure,
• ρ is the fluid density,
• g is the acceleration due to gravity, and
• h is the height of the fluid column.

This equation tells us that the pressure a fluid exerts depends on its density, the gravitational pull, and the height of the fluid column.

If we know the pressure and fluid density, we can rearrange this equation to solve for any variable. For water in the exercise, we used this relationship to find the required height of the water column that would balance the same pressure as the mercury column.
\[ h_{water} = h_{mercury} \times \frac{\rho_{mercury}}{\rho_{water}} \] Because water's density is much less than mercury's, the water column needed is much taller to exert the same pressure. This relationship helps us understand why less dense fluids have higher columns when subjected to the same pressure.