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 \(\left.\rho_{\mathrm{B}}=8.50 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right)\)

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.

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.