Problem 25 - Fundamentals of Physics

Question

In one observation, the column in a mercury barometer (as is shown in Fig. \(14-5 a\) ) has a mcasured height \(h\) of \(740.35 \mathrm{~mm}\). The temperature is \(-5.0^{\circ} \mathrm{C},\) at which temperature the density of mercury \(\rho\) is \(1.3608 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3} .\) The free-fall acceleration \(\mathrm{g}\) at the site of the baromcter is \(9.7835 \mathrm{~m} / \mathrm{s}^{2}\). What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

Step-by-Step Solution

Answer

Atmospheric pressure is about 98988.64 Pa or 742.28 Torr.

Step 1: Understand the Parameters

To determine the atmospheric pressure, we need to use the given parameters: the height of mercury column \(h = 740.35\, \mathrm{mm}\), the temperature \(-5.0^{\circ} \mathrm{C}\), the density of mercury \(\rho = 1.3608 \times 10^{4}\, \mathrm{kg/m^3}\), and the gravitational acceleration \(g = 9.7835\, \mathrm{m/s^2}\).

Step 2: Convert Units of Height

Convert the height of the mercury column from millimeters to meters. Since there are 1000 millimeters in a meter:\[h = \frac{740.35}{1000} \mathrm{~m} = 0.74035 \mathrm{~m}\]

Step 3: Apply the Pressure Formula

The pressure exerted by the mercury column (which equates to the atmospheric pressure) can be calculated using the hydrostatic pressure formula: \[ P = \rho \cdot g \cdot h \]Substitute the given values:\[ P = 1.3608 \times 10^{4}\, \mathrm{kg/m^3} \times 9.7835\, \mathrm{m/s^2} \times 0.74035\, \mathrm{m} \]

Step 4: Calculate Atmospheric Pressure in Pascals

Calculate the atmospheric pressure using the formula:\[ P = 1.3608 \times 10^{4} \times 9.7835 \times 0.74035 = 98988.6374 \mathrm{~Pa} \]Thus, the atmospheric pressure is approximately 98988.64 Pa.

Step 5: Convert Pascals to Torr

To convert from pascals to torr, use the conversion factor:\[ 1 \text{ Torr} = 133.322 \text{ Pa} \]Therefore, \[ \text{Pressure in Torr} = \frac{98988.6374}{133.322} \]

Step 6: Calculate Pressure in Torr

Perform the conversion and calculate:\[ \text{Pressure in Torr} = \frac{98988.6374}{133.322} \approx 742.28 \text{ Torr} \]

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air in the Earth's atmosphere. It is an important concept in fluid mechanics and greatly influences weather patterns and climate. At sea level, standard atmospheric pressure is typically measured as 101,325 pascals (Pa) or 760 torr. However, atmospheric pressure varies depending on altitude and weather conditions.

In everyday terms, atmospheric pressure is what keeps us anchored to the Earth. It affects our daily lives in many ways, such as how our ears "pop" when we change altitude. In scientific terms, measuring atmospheric pressure helps us understand and predict weather conditions. - At sea level, the average atmospheric pressure is about 1013 hPa or 1013 mbar. - Instruments such as barometers are used to measure atmospheric pressure. - Variations in atmospheric pressure can signal weather changes, with low pressure associated with stormy weather and high pressure with clear skies.

Mercury Barometer

A mercury barometer is a classic tool used to measure atmospheric pressure. It consists of a glass tube filled with mercury, inverted in a mercury reservoir. One end of the tube is sealed, while the other end is open and submerged in the mercury reservoir.

As atmospheric pressure increases, it pushes more on the surface of the reservoir, causing mercury to rise in the column. Conversely, if atmospheric pressure decreases, the mercury level in the column falls. This rise and fall of mercury are proportional to changes in atmospheric pressure. The height of the mercury column directly indicates the current atmospheric pressure. - Mercury is used in barometers due to its high density, which makes the device compact. - The unit "torr" is named after Evangelista Torricelli, who invented the mercury barometer. - The use of mercury barometers is slowly decreasing due to the toxicity of mercury, leading to the development of digital barometers.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It is an important concept in fluid mechanics and is used to calculate the force exerted at any given point within a fluid.

The basic formula to calculate hydrostatic pressure is:\[ P = \rho \cdot g \cdot h \]where:- \(P\) is the pressure,- \(\rho\) is the fluid density,- \(g\) is the gravitational acceleration,- \(h\) is the height of the fluid column.

This means that the pressure at a point in a fluid is influenced by its depth, the density of the fluid, and the gravitational force. As you go deeper into the fluid, the pressure increases.- Hydrostatic pressure is essential in understanding how dams can hold back large bodies of water.- It also explains why deep-sea divers need special gear to withstand the water pressure at great depths.- The formula is critical when calculating the pressure inside containers holding liquids.

Unit Conversion

Unit conversion is a key skill in many scientific calculations, allowing comparisons across different measurement systems. Often, calculations require converting units to maintain consistency and accuracy.

In the context of measuring atmospheric pressure, common conversions involve converting between millimeters of mercury (\( mmHg \text{ or } \text{Torr} \) ) and pascals (\( Pa \) ). This requires using specific conversion factors:- 1 \( ext{Torr} = 133.322 ext{ Pa} \)- To convert the height of a column from millimeters to meters, divide by 1000 since there are 1000 millimeters in a meter.
These conversions are vital to ensure accurate and standardized readings across various scientific equipment and measurements applied in different systems.- Consistency in units ensures comparability in experimental data.- It avoids errors that arise from mixing incompatible unit systems.- Understanding unit conversions is crucial when interpreting scientific data from different countries.