Assignment-daixieTM为您提供莱斯特大学University of Leicester PA1120 Light and Matter光与质代写代考和辅导服务！
Instructions:
Light is a form of electromagnetic radiation that travels through space as a wave. It has both particle-like and wave-like properties, and is typically described in terms of its wavelength, frequency, and energy. The wavelength and frequency of light are related by the speed of light, which is constant in a vacuum and is denoted by the symbol c. The energy of light is directly proportional to its frequency, and is given by the equation E = hf, where h is Planck’s constant.
Matter, on the other hand, is composed of atoms and molecules, which are made up of protons, neutrons, and electrons. These particles have mass and occupy space, and interact with each other through various forces, such as electromagnetic forces and the strong and weak nuclear forces. The behavior of matter is governed by the laws of quantum mechanics, which describe how particles move, interact, and are affected by external forces.
In the context of optics, matter can interact with light in a variety of ways, including absorption, reflection, and refraction. When light is absorbed by matter, it can excite electrons to higher energy levels, which can lead to various physical and chemical changes. Reflection occurs when light bounces off a surface, while refraction occurs when light passes through a material and changes direction due to a change in its speed.
A state function for a $\operatorname{Van}$ der Waals gas is given by an equation between thermodynamic variables that depend on model parameters $A, B$, and a physical constant $R$ :
$$
\left(P+\frac{A N^2}{V^2}\right)(V-N B)=N R T
$$
where $A N^2 / V^2$ is referred to as the internal pressure due to the attraction between molecules and $N B$ is an extra volume, sometimes associated with the the volume per molecule.
Write out a differential expression for $d N$ in terms of differentials of the thermodynamic variables.
A state function for a Van der Waals gas is given by an equation between thermodynamic variables that depend on model parameters $A, B$, and a physical constant R:
$$
\left(P+\frac{A N^2}{V^2}\right)(V-N B)=N R T
$$
where $A N^2 / V^2$ is referred to as the internal pressure due to the attraction between molecules and $N B$ is an extra volume, sometimes associated with the volume per molecule.
Write out a differential expression for $d N$ in terms of differentials of the thermodynamic variables.
The solution is pretty straightforward. One way is to differentiate the entire expression and group the terms corresponding to $d N, d P, d T$, and $d V$. Another way to do is by implicit differentiation. The real gas equation can be rewritten such that,
$$
\begin{aligned}
N & =N(T, V, P) \
d N & =\left(\frac{\partial N}{\partial P}\right){T, V} d P+\left(\frac{\partial N}{\partial V}\right){T, P} d V+\left(\frac{\partial N}{\partial T}\right){P, V} d T \end{aligned} $$ In an equivalent way, you could have written the function $P=P(V, T, N)$ and extract $d N$ from the following. $$ d P=\left(\frac{\partial P}{\partial N}\right){T, V} d N+\left(\frac{\partial P}{\partial V}\right){T, N} d V+\left(\frac{\partial P}{\partial T}\right){N, V} d T
$$
For instance, the first term $\left(\frac{\partial P}{\partial N}\right){T, V}$ can be evaluated as $$ \left(\frac{\partial P}{\partial T}\right){P, V}=\frac{N R}{V-N B}
$$
Using any one of the methods you would get
$$
d N=\frac{(V-B N) d P+\left(P-\left(A N^2 / V^2\right)+\left(2 A B N^3 / V^3\right)\right) d V-R N d T}{B P+R T+\left(3 A B N^2 / V^2\right)-(2 A N / V)}
$$
Each day a certain amount of water evaporates from the oceans, lakes, and earth surface and forms water vapor and clouds in the atmosphere. Each day a certain amount of rain falls back to the earth. Make the reasonable assumption that, on the average, the energy consumed by evaporating and lifting the water is equal to the energy produced by condensation and rain falling back to earth.
To evaporate one mole of water, approximately 41090 joules of heat are required and an equivalent amount is expelled when a mole of water condenses.
Estimate the amount of work required each day to produce the rain that raineth. That is, the work to evaporate and lift the water.
You may need to find data to help make this estimate such as the average height of a rain cloud, state what those data are and from whence they came.
Discuss what is supplying this daily energy.
Compare this daily energy consumption with the energy produced each year in metropolitan Boston.
Each day a certain amount of water evaporates from the oceans, lakes, and earth surface and forms water vapor and clouds in the atmosphere. Each day a certain amount of rain falls back to the earth. Make the reasonable assumption that, on the average, the energy consumed by evaporating and liftin the water is equal to the energy produced by condensation and rain falling back to earth. To evaporate one mole of water, approximately 41090 joules of heat
to evaporate and lift the water. You may need to find dat a to help make this estimate such as the average height of a rain cloud, state what those data are and from whence they came. Discuss what is supplying this daily energy. Compare this daily energy consumption with the energy produced each year in metropolitan Boston.
From the problem statement,
The energy to lift water $=$ energy produced by condensation and rain
The first step would be to find the volume of water evaporated in the surface of earth. Since all the water is assumed to precipitate and et urned as rain, an average rainfall data and earth surface area would be needed to estimate this. A simplifying assumption for the calculation of volume of water precipitated could be that the curvature of surface is very small compared to the average height of precipitation. (The following data are from problem solutions by the group Ryan Clark, Daniel MacPhee, Taylor Schildgen, Alex Bradley – reported taken from climate prediction center, 1979-2000 dat a).
global precipitation average per day $=3 \mathrm{~mm}$
radius of earth $=6371 \mathrm{kms}$
surface area of earth $=5.1^* 10^{14} \mathrm{~m}^2$
volume of precipitation $=$ surface area $$ average precipitation $=3.0^ 10^{-3} * 5.1 * 10^{14} \mathrm{~m}^3=$ $15.3^* 10^{12} \mathrm{~m}^3$
mass of precipitation $=$ density $$ volume $=15.3^ 10^{15} \mathrm{~kg}$ (density of water $\left.=1 \mathrm{gm} / \mathrm{cc}\right)$
number of moles of water $=$ total mass $/$ molar mass $=15.3 / 18^* 10^{15}=8.5^* 10^{14}$
Next step is to calculate the energy spent on lifting this estimated volume of water to the atmosphere. The energy needed to evaporate water is given in the problem statement as 41090 joules per gram. you would also need some information on the average height these drops are lifted or the average cloud height to calculate its potential energy change. The total energy per day day can be estimated from these calculations. This amount of energy is consumed everyday by the water that rainet $h$.
A reasonable distance at which clouds are formed could be around $5000 \mathrm{kms}$.
The energy needed to lift the mass of water to this height $=\mathrm{m}^* \mathrm{~g}^* \mathrm{~h}=15.3^* 10^{15} * 9.8^$ 5000 Joules $=7.497^ 10^{20}$ joules
The energy needed to evaporate the entire volume of water $=$ number of moles $*$ energy for evaporation per mole $=8.5 * 10^{14} * 41090$ Joules $=3.49265 * 10^{19}$ Joules.
The total energy to evaporate and lift $=(7.497+.85)^* 10^{20}$ Joules $=8.347^* 10^{20}$ Joules.
Discussion on what is supplying the energy to lift this amount of water can be on fraction of sun’s energy reaching earth, on the fraction of water surface available, average global temperature, ozone layers and anything else that might affect the supply of energy to evaporate and lift water vaopur.
There are four state variables for an arbitrary pure gas: total moles of gas molecules, pressure, volume, and temperature.
How many independent state variables exist for a mole of an ideal gas?
Using different combinations of independent variables, write as many unique (but numerically equivalent) expressions for the molar internal energy of an ideal gas as possible.
One mole of an ideal gas has 2 state variables. The four degrees of freedom (n, $P, V$, and $T$ ) for an arbitrary gas are decreased by the constraints, $n=1$ and $P V=n R T$.
We can write $\bar{U}$ in terms of each combination of two independent variables: $\bar{U}(T, \bar{V}), \bar{U}(T, P), \bar{U}(P, \bar{V})$. The differential form for the molar internal energy of an ideal gas is given by
$$
d \bar{U}=\overline{C_v} d T
$$
Integration gives:
$$
\bar{U}(T)=C_v\left(T-T_0\right)=U(T, \bar{V})=U(T, P)
$$
Substitution using the ideal gas law gives:
$$
\overline{U(P, \bar{V})}=\frac{\overline{C_v}}{R}\left(\bar{V} P-\bar{V}_0 P_0\right)
$$
