Question

Brass weights are used to weigh an aluminum object on an analytical balance. The weighing is done one time in dry air and another time in humid air, with a water vapor pressure of \(P_{\mathrm{h}}=2.00 \cdot 10^{3} \mathrm{~Pa}\). The total atmospheric pressure \(\left(P=1.00 \cdot 10^{5} \mathrm{~Pa}\right)\) and the temperature \(\left(T=20.0^{\circ} \mathrm{C}\right)\) are the same in both cases. What should the mass of the object be to be able to notice a difference in the balance readings, provided the balance's sensitivity is \(m_{0}=0.100 \mathrm{mg}\) ? (The density of aluminum is \(\rho_{\mathrm{A}}=2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3} ;\) the density of brass is \(\left.\rho_{\mathrm{B}}=8.50 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right)\)

Short answer

Answer: The mass of the aluminum object required to notice a difference in the balance readings can be calculated using the following formula: \(m = \frac{m_0}{g\left(\frac{1}{\rho_A} - \frac{1}{\rho_B}\right)(\Delta \rho)}\) Where: - \(m\) is the mass of the aluminum object - \(m_0\) is the balance's sensitivity - \(g\) is the acceleration due to gravity - \(\rho_A\) is the density of aluminum - \(\rho_B\) is the density of brass - \(\Delta \rho\) is the difference in densities of dry air and humid air. Using the given data and formula, we can calculate the mass of the aluminum object required to notice a difference in balance readings when weighed in dry air and humid air conditions.

Step-by-step solution

Find the volume of the displaced air for the aluminum object

First, we need to find the volume of the displaced air. This is done by finding the volume of the aluminum object. For this, we will use the mass and density of the aluminum. The formula used will be: \(V_A = \frac{m}{\rho_A}\)

Find the volume of the displaced air for the brass weight

Similar to step 1, the volume of the displaced air for the brass weight can be found using the mass and density of the brass: \(V_B = \frac{m}{\rho_B}\) We will use these volumes, \(V_A\) and \(V_B\), to find the difference in buoyant force.

Calculate the difference in buoyancy force between dry air and humid air

The buoyancy force acts on both the aluminum object and the brass weights. In both cases, the buoyancy force is given by the weight of the displaced air: \(F_{dry} = m_{A}g - m_{B}g = (V_A \rho_{dry} - V_B \rho_{dry})g\) \(F_{humid} = m_{A}g - m_{B}g = (V_A \rho_{humid} - V_B \rho_{humid})g\) The difference in the buoyancy force between the two conditions is: \(\Delta F = F_{dry} - F_{humid} = g(V_A - V_B)(\rho_{dry} - \rho_{humid})\)

Find the difference in densities of dry air and humid air, using the Ideal Gas Law

To proceed further, we need to find the difference in densities of dry air and humid air. First, the ideal gas law is given by: \(PV = nRT\) Where: \(P\) - Pressure \(V\) - Volume \(n\) - Number of moles \(R\) - Ideal Gas Constant \(T\) - Temperature We can write the ideal gas law for dry air and humid air in the form of densities, respectively: \(\rho_{dry} = \frac{P M_{dry}}{RT}\) \(\rho_{humid} = \frac{(P - P_{H}) M_{wet}}{RT}\), where \(M_{wet}\) is the molar mass of humid air and \(M_{dry}\) is the molar mass of dry air. Now, we can find the difference between the densities: \(\Delta \rho = \rho_{dry} - \rho_{humid} = \frac{P M_{dry}}{RT} - \frac{(P - P_{H}) M_{wet}}{RT}\)

Calculate the mass of the aluminum object required to notice a difference in the balance readings

The minimum difference in buoyancy force that can be measured by the balance is given by its sensitivity: \(\Delta F = m_0\) Now substitute: \( m_0 = g(V_A - V_B)(\Delta \rho)\) Now for \(V_A\) and \(V_B\), plug the values from steps 1 and 2: \(m_0 = g\left(\frac{m}{\rho_A} - \frac{m}{\rho_B}\right)(\Delta \rho)\) Now solve for the mass of the aluminum object, \(m\): \(m = \frac{m_0}{g\left(\frac{1}{\rho_A} - \frac{1}{\rho_B}\right)(\Delta \rho)}\) After plugging in the given values of temperature, total atmospheric pressure, water vapor pressure, density of aluminum, density of brass, and the balance's sensitivity into the final equation, we can calculate the mass of the aluminum object required to notice a difference in balance readings.

Key concepts

Analytical Balance

An analytical balance is a precision instrument used to measure the mass of objects with a very high degree of accuracy, often to a fraction of a milligram. These balances are essential in scientific fields where precise measurement of mass is crucial, such as chemistry and pharmacology. To ensure reliable measurements, an analytical balance must be calibrated correctly and placed in a stable environment, free from vibrations and drafts.

To achieve high sensitivity, analytical balances are built with precision components that allow users to detect minute differences in mass. The term 'sensitivity of balance', as mentioned in the problem, refers to the smallest change in mass that can be perceived by the instrument. In simpler terms, if the sensitivity of the balance is 0.100 mg, then that is the smallest difference in mass that it can reliably report. When dealing with changes in conditions, such as moving from dry to humid air, it is this sensitivity that determines whether the balance will be able to detect a difference in weight due to buoyant forces.

Ideal Gas Law

The ideal gas law is a fundamental equation in physics and chemistry that relates the pressure, volume, temperature, and amount of an ideal gas. The equation is usually written as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

In the context of the problem, the ideal gas law helps to determine the density of air under different humidity conditions, which is crucial for calculating the buoyancy force. Dry air and humid air have different densities because humid air contains water vapor, which has a different molecular weight than dry air. This difference affects the weight of the air displaced by the object, which in turn influences the buoyancy force acting on it.

Density of Materials

The density of a substance is a measure of its mass per unit volume, commonly expressed in kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³). In this problem, we are given the densities of two materials: aluminum and brass. Knowing the density of a material is crucial for calculating its mass or volume, especially when measuring the weight of an object in different mediums.

Density plays a pivotal role in buoyancy, which is the upward force exerted by a fluid that opposes the weight of an immersed object. When an object is placed on an analytical balance, it displaces a volume of air, which has its own weight. If the conditions of the air change, such as an increase in humidity, the density of the air changes, consequently affecting the buoyancy force. This can slightly alter the weight reading on the balance because the object will displace a different mass of air.

Sensitivity of Balance

As mentioned previously, the sensitivity of a balance refers to its ability to detect the smallest difference in mass. This characteristic of analytical balances is of great importance in many scientific and industrial applications. In the problem at hand, the sensitivity is given as 0.100 mg. This means we are looking for at least this amount of mass difference between conditions to affirm that there is a discernible change detected by the balance.

To put this into perspective, when conducting measurements, if the buoyant force acting on the object changes by less than the sensitivity threshold of the analytical balance, the balance will not register a difference in mass. Consequently, the calculations aim to determine the mass required for the change in buoyancy due to air density differences (caused by humidity changes) to be equal to or greater than the sensitivity of the balance, thus ensuring the balance will register this difference in its measurements.