Question

Evaluate the translational partition function for Ar confined to a volume of \(1000 . \mathrm{cm}^{3}\) at \(298 \mathrm{K}\). At what temperature will the translational partition function of Ne be identical to that of Ar at \(298 \mathrm{K}\) confined to the same volume?

Short answer

From the given information, we first calculate the mass of one Ar and one Ne atom using their molar mass. Then, we calculate the translational partition function for Ar at 298 K and a volume of 1000 cm³ using the formula \(q_{trans} = \frac{V(2\pi mkT)^{3/2}}{h^3}\). Next, we set the translational partition functions of Ar and Ne equal to each other and solve for the unknown temperature (T) at which Ne's translational partition function will be equal to Ar's translational partition function at 298 K. The final equation to solve for T is \( T = \frac{(m_{Ar})(298)}{m_{Ne}} \), where \(m_{Ar}\) and \(m_{Ne}\) are the masses of Ar and Ne atoms calculated earlier.

Step-by-step solution

First, we need to calculate the mass of one Ar and one Ne atom using their molar mass. Molar mass of Ar: \(39.95 \frac{g}{mol}\) Molar mass of Ne: \(20.18 \frac{g}{mol}\) To convert molar mass to the mass of one atom, we need to divide by Avogadro's number (NA), which is approximately \(6.022\times10^{23}\) particles/mol. Mass of one Ar atom: \(m_{Ar} = \frac{39.95 \frac{g}{mol}}{6.022\times10^{23} particles/mol} \) Mass of one Ne atom: \(m_{Ne} = \frac{20.18 \frac{g}{mol}}{6.022\times10^{23} particles/mol} \) #Step 2: Calculate the translational partition function for Ar at 298 K#

Now that we have the mass of one Ar atom, we can calculate its translational partition function at 298 K and a volume of 1000 cm³ using the given formula: \(q_{trans_{Ar}} = \frac{1000 (2\pi m_{Ar}k(298))^{\frac{3}{2}}}{h^3}\) where: - h is Planck's constant (\(6.626\times10^{-34} Js\)) - k is Boltzmann's constant (\(1.381\times10^{-23} J/K\)) - \(m_{Ar}\) is the mass of one Ar atom, calculated in Step 1. #Step 3: Set the translational partition functions of Ar and Ne equal to solve for Ne's temperature#

Now we want to find the temperature at which the translational partition function of Ne will be identical to that of Ar at 298 K. To do this, we will set the partition functions of Ar and Ne equal to each other and solve for the unknown temperature, T. \(q_{trans_{Ar}}(298) = \frac{1000 (2\pi m_{Ar}k(298))^{\frac{3}{2}}}{h^3} \) \(q_{trans_{Ne}}(T) = \frac{1000 (2\pi m_{Ne}k(T))^{\frac{3}{2}}}{h^3} \) \(q_{trans_{Ar}}(298) = q_{trans_{Ne}}(T) \) After equating the partition functions of Ar and Ne, solving for T (temperature of Ne) is the next step. #Step 4: Solve for Temperature (T)#

Now, we have an equation in terms of T, the unknown temperature at which Ne's translational partition function will be equal to Ar's translational partition function at 298 K: \(\frac{1000 (2\pi m_{Ar}k(298))^{\frac{3}{2}}}{h^3} = \frac{1000 (2\pi m_{Ne}k(T))^{\frac{3}{2}}}{h^3} \) We can simplify it by canceling equal quantities on both sides: \((2\pi m_{Ar}(298))^{\frac{3}{2}} = (2\pi m_{Ne}(T))^{\frac{3}{2}} \) Now raise both sides to the power \(\frac{2}{3}\) to get rid of the fractional exponent: \( (2\pi m_{Ar}(298)) = (2\pi m_{Ne}(T)) \) Next, divide both sides by \(2\pi m_{Ne}\): \( \frac{(2\pi m_{Ar}(298))}{(2\pi m_{Ne})} = T \) Finally, substitute the values of \(m_{Ar}\) and \(m_{Ne}\) calculated in Step 1, and solve for T: \( T = \frac{(m_{Ar})(298)}{m_{Ne}} \)

Key concepts

Thermal Physics

Thermal physics is a fascinating branch of physics that deals with the study of heat energy and its transformation to other forms of energy. It plays a critical role in understanding how energy is exchanged and utilized in our everyday environment. In particular, thermal physics is pivotal in describing how particles, like gas atoms, move at different temperatures.

When considering translational motion in gases, thermal physics helps to determine the behavior of gas particles influenced by changes in temperature and volume. This is especially important in the case of the translational partition function, which quantifies the distribution of a gas particle's energy among its different degrees of freedom.

Avogadro's Number

Avogadro's Number, denoted as \(N_A\), is a fundamental constant in chemistry and physics. It represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance.

It is approximately \(6.022 \times 10^{23}\) particles/mol. Avogadro's Number is vital for calculating the mass of an individual atom or molecule based on the molar mass given in grams per mole.

For instance, when determining the mass of a single atom in gases such as argon (Ar) or neon (Ne), we divide their respective molar masses by Avogadro's Number. This process is crucial to evaluate properties like the translational partition function.

Boltzmann's Constant

Boltzmann's constant, represented by \(k\), serves as a bridge between macroscopic and microscopic physics. It is a physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas.

The value of Boltzmann's constant is approximately \(1.381 \times 10^{-23} \) J/K, playing an integral role in statistical mechanics and thermodynamics.

In the calculation of the translational partition function, Boltzmann's constant is used to associate temperature with the energy states of a particle, allowing us to predict the behavior of gases at various temperatures.

Planck's Constant

Planck's constant is a fundamental constant denoted by \(h\), crucial in quantum mechanics. It quantifies the size of action in the quantum realm and is approximately \(6.626 \times 10^{-34} \, \mathrm{Js}\).

This constant is vital for understanding the particle-wave duality of light and matter. In the context of the translational partition function, Planck's constant helps in quantifying the available energy states a particle can occupy.

While calculating the translational partition function, we use Planck’s constant to derive the density of states which relates to the behavior of subatomic particles within a defined volume at given temperatures.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation of state in chemistry and physics, expressed in the formula \(PV = nRT\). It describes how the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of a gas relate to the amount of gas measured in moles (\(n\)), with \(R\) as the ideal gas constant.

This law provides a good approximation of the behavior of many gases under various conditions, assuming that the gas particles have no interactions and occupy no volume themselves.

For calculating the translational partition function, the ideal gas law assists in relating macroscopic properties to microscopic behaviors by providing context about the conditions under which gases behave ideally. This understanding is key to predicting and calculating properties like pressure, temperature, and volume interactions in different gaseous systems.