An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.
Under various conditions of temperature and pressure, many real gases behave qualitatively like an ideal gas where the gas molecules (or atoms for monatomic gas) play the role of the ideal particles. Many gases such as nitrogen, oxygen, hydrogen, noble gases, some heavier gases like carbon dioxide and mixtures such as air, can be treated as ideal gases within reasonable tolerances over a considerable parameter range around standard temperature and pressure. Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure, as the potential energy due to intermolecular forces becomes less significant compared with the particles' kinetic energy, and the size of the molecules becomes less significant compared to the empty space between them. One mole of an ideal gas has a volume of 22.710947(13) litres at standard temperature and pressure (a temperature of 273.15 K and an absolute pressure of exactly 105 Pa) as defined by IUPAC since 1982.The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size becomes important. It also fails for most heavy gases, such as many refrigerants, and for gases with strong intermolecular forces, notably water vapor. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point of low temperature and high pressure, real gases undergo a phase transition, such as to a liquid or a solid. The model of an ideal gas, however, does not describe or allow phase transitions. These must be modeled by more complex equations of state. The deviation from the ideal gas behavior can be described by a dimensionless quantity, the compressibility factor, Z.
The ideal gas model has been explored in both the Newtonian dynamics (as in "kinetic theory") and in quantum mechanics (as a "gas in a box"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the Drude model and the free electron model), and it is one of the most important models in statistical mechanics.
If the pressure of an ideal gas is reduced in a throttling process the temperature of the gas does not change. (If the pressure of a real gas is reduced in a throttling process, its temperature either falls or rises, depending on whether its Joule–Thomson coefficient is positive or negative.)
The book I am reading says that by definition, the ideal gas satisfies the equations
$$PV=nRT\tag{1}$$
$$\left (\frac{\partial U}{\partial P}\right )_T = 0\tag{2}$$
where does (2) come from? In other words, what justifies this equation in the definition above?
Consider ##n## moles of a gas at a constant temperature ##T##.
If we vary pressure ##P## and measure the corresponding values of volume ##V##, we can make a plot of ##P\frac{V}{n}=Pv## against ##P##.
This gives us some graph which has some form. Turns out that for a range of pressure starting...
I dont have an solution Attempt. Maybe something with PV=nRT but this is for ideal gas and H2O is liquid. An other formula they introduced us to is: dE=-P*V
Last month @Chestermiller opened the thread: Focus Problem for Entropy Change in Irreversible Adiabatic Process.
I couldn't wrap my head around something apparently simple but the thread was not about that so I was instructed to open a new thread to discuss it separately and keep the original...
In his Chapter 13.3 (2nd edition), Callen gives the standard form for the virial expansion for the mechanical equation of state of a fluid as an exapnsion in powers of the molar volume ##v##:
$$P = \frac{RT}{v}\left(1 + \frac{B(T)}{v} + \frac{C(T)}{v^2} + \dots \right) \equiv P_{ideal} +...
Callen asks us (with respect to an ideal gas)
I had thought to proceed as follow. We have the definition for the singular reaction:
$$\ln K_s(T) = - \sum_j \nu_j \phi_j(T).$$
Now a reaction which is the sum of this reaction with itself (doubled reaction) has ##\nu_j \to 2\nu_j## so that its...
In Ch. 13.1 of the second edition, Callen defines a general ideal gas as follows:
Of course, all of these can be proved as a theorem of statistical mechanics given a no-interaction assumption.
At any rate, my claim is about Callen's claim that a single component ##j## of general ideal gas...
In Chapter 13.2 of his text, Callen states that the chemical potential with respect to the ##j##th component of an ideal gas can be written as
$$\mu_j = RT \left[\phi_j(T) + \ln P + \ln x_j \right].$$
He states this outright and doesn't prove it, and I am trying to do so now. Based on what has...
I have an ideal gas in a cylinder with a massless, frictionless piston, and the gas starts out at To, Po, Vo. The system is adiabatic. Initially, the gas is in equilibrium with an external pressure, also at Po. I initiate an irreversible process by instantly dropping the external pressure to P1...
This is my first post of a homework problem, and I am just trying to make sure I am not missing anything as I help a child who is taking a college class on this stuff this summer. And to be clear, just trying to help her understand stuff, not help her with the homework problem. So I won't go...
The text derives C_p-C_v=nR for ideal gasses. They start with $$H = U + PV = U + nRT$$ for ideal gas. Since U is only a function of temperature for an ideal gas, the right-hand side is only a function of temperature so $$\frac{dH}{dT} = \frac{dU}{dT} + nR$$. Now the text does something I...
In his classic textbook, Callen remarks that
I have labelled the claims (1) and (2). I am not sure about either. For the first, I have tried to proceed as follows (all equations are from Callen's second edition and all 0 subscripts are with respect to some reference state of an ideal gas):
I...
For this problem,
The solution is,
However, why must we use absolute temperature for the ideal gas law (i.e why can we not use Celsius for T)
Many thanks!
TL;DR Summary: Struggling to structure the problem and derive an analytical solution for gas expanding into other gas in a rigid tank. Preferred formulation is fixed control volumes. This is not a homework problem.
The problem:
Two control volumes (A and B) are in a rigid tank filled with air...
Question: Two samples of a monatomic ideal gas are in separate containers at the same conditions of pressure, volume, and temperature (V = 1.00 L and P = 1.00 atm). Both samples undergo changes in conditions and finish with V = 2.00 L and P = 2.00 atm. However, in the first sample, the volume is...
To solve this problem I used two equations:
$$
PV=nRT,
$$
where ##P## is the pressure, ##V##the volume, ##R##the gas constant, ##T##for temperature and is##n##the number of moles, related to the
mass ##m## and molar mass ##M## by
$$
n=\frac{m}{M}.
$$
It will be also necessary consider the...
The answer given for part (c) in the back is that temperature doesn't change as the gas in cylinder A expands to fill cylinder B.
The thermodynamic system here is composed of the two cylinders A and B joined by some pipe.
But, I cannot find a satisfactory explanation for temperature...
There is no atmosphere pressure.
My work :
pA=k(x-x0) => pA=(k/A)(V-V0)
But this should be false beccause I want to use W=∫PdV to find work done by the gas but my final anwer is wrong ...
Please guide me where my mistake is if you have enough time. Thanks.
Homework Statement:: I am trying to understand a formula given in our book for determining molar heat capacity of an ideal gas under different thermodynamic processes using a single formula, but it is confusing. The exact formula for different processes is in the screenshots below. Can someone...
1.Does the Maxwell Boltzmann distribution change depending on the shape of the container? Pressure and the volume is constant. How is the Distribution affected whether the gas is in: a,sphere b,cube c,cuboid?
Why does/doesn’t the distribution change depending on the shape of the container...
P1=1.9 and P2=4.8.
Question: what is the total change in internal energy
This is what I have so far but it is still incorrect I believe:
U= (3/2)(1/2)(2.9)(1.01x10^3)(8x10^-3)
Where am I going wrong?
For now it is only about the 1 task
If the task states that:
You can approximate that their dynamics in water resembles that of an ideal gas.
Does it then mean that I can take glucose as the ideal gas and then simply calculate the entropy for the ideal gas?
For a freely expanding ideal gas(irreversible transformation), the change in entropy is the same as in a reversible transformation with the same initial and final states. I don't quite understand why this is true, since Clausius' theorm only has this corrolary when the two transformations are...
(The equation of ideal gas is PV=NRT.if P=1atm,N=1mole,T=0°K,R=gas constant then volume = zero. Hence, the volume of an individual molecule of ideal gas is zero)
An individual molecule of ideal gas is assumed to have zero volume. The molecules of ideal gas are assumed to be dimensionless points...
Hi! I wanted to do some basic calculations for temperature T on a water-filled pot. I noticed something strange on my calculations, and couldn’t figure out what was wrong...
So here it is:
The ideal gas formula:
k=PV
The actual formula Relates equally the product PV with the a constant...
Question:
Answer:
In the third last line of working, I do not understand why the pressure variable is changing? Shouldn't pressure remain constant and only the Volume change?
Hello.
Firstly, I've calculated the density of Kr ( = 3.74 g/dm3), and I know that the p (fluid) = ρ * h * g. And then I've used the following equation: p1*V1 = p2*V2, and therefore: p1*V1 = ρ * h * g * (m/ρ) => p1*V1 = h * g * m. (h = 3.0153 m) Is that correct? Please, how could I calculate...
a) We use the definition of heat transfer in a gas at constant volume:
Q = n*C_v*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 4.99 J
b) We use the definition of heat transfer in a gas at constant pressure:
Q = n*C_p*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 8.31 J
c) In both processes delta_U...
So for a collection of particles each with mass m, the pressure beneath them, ##p(z)## should be higher than the pressure above them ##p(z + \Delta z)##.
This is a change in force per unit area (force per unit volume I suppose) times a volume to equate with the gravitational force
$$ \frac...
Summary:: Heat capacity for real gas with ideal gas (zero pressure) equation
I'm looking at this problem and I'm stuck.
I usually question everything but this problem is confusing me.
I don't know how they've made the jump from reduced properties (from generalized Cp charts(?)) to...
Summary:: Gibbs and Helmholtz energies calculations for heating an ideal gas at constant volume
I am solving a problem involving an ideal gas that undergoes several chained changes of state. One of the steps asks to calculate the change in Gibbs Energy (DeltaG) and Helmholtz energy (Delta A)...
My teacher likes to make really weird problems. How can I start this problem? The only thing I thought of doing is using the polytropic ideal gas equation when cp= constant. (p2/p1)^k-1/k = T2/T1 and making p1 and t1 in each case the normal state of the lungs
Hey there! for this problem i try to use the combinate gas ecuation. First of all the values its necesary to have it in absolutes:
70 F = 527.67 K
90 F = 549.67 K
The ecuation looks like: (200 psig) (1 ft^3)/529.67 K = (0.3 InHg) V2/549.67 K I can eliminate "K" but not psig with InHg for obtain...
It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.
How can we find a equation of a 1D sound wave in a non-differential form in an ideal gas with viscosity? How does the damping work? How does the wave lose energy at each layer as it propagates?
To be clear I am looking for a simple exponential-sinusoidal function for it just in the case of...
We know that
$$dU=\delta Q + \delta W$$
$$dU = TdS - pdV$$
So from this:
$$dS = \frac{1}{T}dU + \frac{1}{T}pdV \ (*)$$
For an ideal gas:
$$dU = \frac{3}{2}nkdT$$
Plugging that into (*) and also from ##p=\frac{nRT}{V}## we get:
$$S = \frac{3}{2}nk \int^{T_2}_{T_1} \frac{1}{T}dT +...
My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$
Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log...
Do particles have air in between them in the ideal gas model?
I think the answer is 'no, but I am not quite sure about the explanation. Is it because in an ideal gas model, the volume of the particles is negligible? Thank you.
a. The piston will be at rest when all its kinetic energy converted into work to push the gas, so:
$$\frac{1}{2}m_0 c^2=P_0. \Delta V$$
$$\frac{1}{2}m_0 \frac{29}{4} \frac{P_0.V_0}{m_0}=P_0.\Delta V$$
$$\frac{29}{8} V_0=\Delta V$$
$$\frac{29}{8} L_0 = L_0 - L$$
$$L=-\frac{21}{8} L_0$$
My...
It even gives a hint, it says "consider two horizontal surfaces z1 and z2 and think about what thermodynamic equilibrium means for particles traveling from one surface to the other". This really trips me up because I am not sure what to do with this. Obviously in equilibrium the number of...
I have come up with the change in height as 170 cm. My professor does not want to solve for the problem for a reason I do not understand. 170 cm is not part of the answer key. The answer according to the answer key is 65 cm.
My attempt is:
Initial temperature:
p=F/A; (50 *9.8) / (pi * 0.05^2)...
I figured that T' is a common factor for both relationships and from there deduceted that T'=p2xt1/p1=v1xt2/v2. However, I don't understand how that can be further manipulated to PV=KT.
Is this right for difference between idea gas and perfect gas. trying to get it into head but can't find simple explation.Idea gas
it is a fictious matter that follows the PV=nRuT or PV = mRuT equation, which has predermined conditions of ideal conditions of the the gas. As temperature for...
Solution from the textbook:
My work:
I constantly get 1.55kg. I also tried dividing the first and the second equation (pxV=m/M x R x T with different values). How did they come up with the equation in the solution? Also, I am sorry if I posted it in the wrong place and didn't follow the rules...
A cylinder contains an initial volume V1 = 1m^^3 of a perfect gas at initial pressure p1 = 1 bar, confined by a piston that is held in place by a spring. The gas is heated until its volume is doubled and the final pressure is 5 bar. Assuming that the mass of the piston is negligible and that the...
Hi,
I didn't understand the maths involved in the below article in regard to temperature and ideal gas thermometer. If any member knows it, may reply me.
If triple point of water is fixed at 273.16 K, and experiments show that freezing point of air-saturated water is 273.15 K at 1 atm...
Hi,
I tried to do this question in two different approaches one of them was using the equation PV=mRT where I got the right answer which is 4.305 m**2. However, I tried using this Density = Mass/Volume, where I substituted Denisity= 1.225 and Mass equals 5kg to get the volume as 4.08.
Can...