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Animation of Thermometer Using Expanding Gas as Sensitive Medium Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
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Thermometer Using Expanding Gas

Objective of this Animation

This animation's objective is to show how a gas can be heated and how its temperature is increased and how that temperature change can be used as a thermometer. An advantage of a gas for a thermometer is that gases have fewer phase changes over wide temperature ranges. The best gas for a gas thermometer is helium because it has no phase changes from its liquefaction point, 4.2K, upward. The average speed of the gas atoms is increased by gas atom collisions with the vibrating atoms in a solid plate that is interfaced below the gas atom reservior. The gas atoms with increased speed place more pressure (force) on a spring-loaded movable piston and cause it to increase the volume of the gas reservoir. The increase in Kelvin (absolute) temperature is proportional to the increase in gas pressure and gas volume.

Setting the Temperature Scale

The simplest way to show the effects of increasing temperature is to start with a temperature of 0 K so that the gas atoms are not moving at all. Then add heat by exciting the atoms in the metal plate so that they bump into the gas atoms and increase their speed and energy. The moving gas atoms apply a force to the piston and accelerate it as well as compress the spring and increase its potential energy. When the piston moves to its new position, the frictional force causes energy loss as well. Then, after the piston comes to rest, the total energy change is

`deltaE_("total")=deltaE_("spring")+deltaE_("friction")+deltaE_("gas")`

Having chosen the zero value, we then need to calibrate the scale with some physically accessible phenomena. The standard K values used are the freezing value of water at 273.15 K and the boiling point of water under earth's standerd atmospheric pressure, 373.15 K.

Animation of Thermometer Using Expanding Gas as Sensitive Medium

Note:The Temperature Starts at 0 Kelvin so There Can Be No Animation:

Please Press "Start Heating" to Start the Animation

↓

Macroscopic Physics of the Animation

In this animation, the motion of atoms in the bottom plate is increased and these hotter atoms interact with the gas atoms above the plate causing their temperature (translational energy) to become larger. The heated gas atoms apply more force (pressure) to the piston and the piston compresses the spring to a new setting which gives a linear readout of the new gas temperature.

For an ideal monoatomic gas the macroscopic equation for gas pressure, `P`, that pushes on the piston is

`P=Nk_BT/V`

where `N` is the total number of atoms, `V` is the volume of the container, `k_B` is Boltzmann's constant and `T` is the Kelvin temperature of the gas. The total energy of that gas is just

`E_("thermal")=3/2Nk_BT`

The program computes the total thermal energy, `E_("thermal)"`, of the gas which is the same as `3/2Nk_BT`.

`P=(2/3)E_("thermal")/V`

Since some of the gas energy is transferred to the spring attached to the piston, if we know the displacement of the piston from startup, we will have a linear readout of `E_("thermal)"`: p style='text-align: center;'> `Nk_BdeltaT=3/2k_("spring")(x_P-x0_P)`

If we know the initial temperature, `T_0`, then we can define the temperature as p style='text-align: center;'> `deltaT=[3/2k_("spring")]/(Nk_B)(x_P-x0_P)`

Differential Equation for Motion of Piston

The equation for the position,`x_P`, of the piston is

`m_P(d^2x_P)/(dt^2)+b(dx_P)/dt+k_("spring")(x_P-x0_P)=F(t)`

where `m_P` is the piston mass, b is a piston drag coefficient, k is a coefficient of spring force per unit compression displacement, and F(t) is the force applied by the gas atoms to the left side of the piston. To avoid extra oscillations of the spring-mass system, we will generally use critical damping where

`b=2sqrt(k_("spring")m_P)`

In order to show the difference of motion of critical damped and less than critical damping I have included a slider with which one can set a damping ratio somewhere between zero and critical damping. This allows the learner to see the energy exchange between the gas and the spring compression as the piston moves. Of course, this should be done when there is no heat being applied to the gas.

Since we don't have a functional form for F(t), we will need to solve this equation by iteration. We will use Euler's method to solve for x and v and for each `F[t]` as it comes.

First we will rename the first derivative of x as

`v=dx/dt`.

Then the second differential equation becomes

`(dv)/dt=(F[t]-kx-bv)/m_P`

Now let a single time step be `deltat=h`. Then the next iteration of the two equations is

`x_(i+1)=x_i+hv_i`
`v_(i+1)=v_i+h(F_i-kx_i-bv_i)/m_P`
`x_(i+2)=x_i+hv_(i+1)`
`v_(i+2)=v_(i+1)+h(F_(i+1)-kx_(i+1)-bv_(i+1))/m_P`

Energy loss due to drag.

At any time the repelling force, `F_d`, due to drag is

`F_d=bv(t)`

Of course the energy lost per frame, `deltaE_d`, due to the drag force is that force multiplied by the distance, `delta_x`, traveled during during the frame:

`deltaE_d=bv(t)delta_x`

But `delta_x` during the frame is just `hv(t)` so we have

`deltaE_d=bhv(t)^2`

Then we just have to sum all increments of `deltaE_d` to get the total energy lost to drag.

Appendix: Details of Microscopic Particle Energy Transfer

How Energy is Transferred from Metal Plate to Gas

The 3D velocities of both gas atoms and solid atoms are needed. We will assume that the mass, `M`, of the solid atom is infinite with respect to the gas atom mass, `m`. Then, during collisions, the reduced mass of that pair is just `2m` and the gas cannot heat the solid. If we let the solid atoms be represented by `A2` and the gas atoms be represemted by `A1` and name the 3D differences in postion

`[deltax,deltay,deltaz]=[A2.x-A1.x,A2.y-A1.y,A2.z-A1.z]`

and name the distance just before collision `d`.

`d=sqrt[deltax^2+deltay^2+deltaz^2)]`

then the following equations will compute the 3D velocity of the gas atom after the collision:

First we need the unit vector along line between centers.

`U=[ux,uy,uz]=[(deltaz)/d,(deltay)/d,(deltaz)/d]`

To obtain the energy change after the collision we need the initial energy, `E_i`, of the gas atom:

`E_i=m/2((A_1.v_x)^2+(A_1.v_y)^2+(A_1.v_z^2))`

We also need the difference in 3D velocity:

`deltaV=[deltav_x,deltav_y,deltav_z]=[A2.v_x-A1.v_x,A2.v_y-A1.v_y,A2.v_z-A1.v_z]`

Then we take the dot (inner) product of unit vector with difference in 3D velocity and multiply by 2m

`2mUdeltaV=2m_1[ux,uy,uz]*[deltav_x,deltav_y,deltav_z]`

Finally we have the 3D velocity CHANGE components of A1:

`deltavx_1=2m_1u_x(deltav_x)/m_1,deltavy1=2m_1u_y(deltav_y)/m_1,dvz1=2m_1u_z(deltav_z)/m_1;`
`deltavx_1=2u_xdeltav_x,deltavy1=2u_ydeltav_y,dvz1=2u_zdeltav_z`

Finally velocity of A1 after collision:

`A1.vx+=deltavx_1,A1.vy+=deltavy_1,A1.vz+=deltavz_1`

Then the final energy is

`E_f=m/2[(A_1.v_x+deltavx_1)^2+(A_1.v_y+deltavy_1)^2+(A_1.v_z^2_deltavz_1)]`

Note that the values of `[deltavx_1,deltavy_1,deltav_z1 ]` are just as likely to be negative or positive but, on average, energy will be transferred to the gas atom because of their squares `[deltavx_1^2,deltavy_1^2,deltav_z1^2]`

Motion of Gas Atoms with Respect to Cylinder Boundaries and Piston

When the atom hits the top or left or bottom boundary of the container the normal component, `v_n`, of its velocity is negated:

`bbv_(out)=bbv_("in")-2bbhatn*bbv_("in")`

where `bbhatn` is the normal to the boundary.

When the gas atom, moving at `bbV_(in)=[v_x,v_y,v_z]`, hits the piston which is moving along the x axis at [`vp_x`,0,0] its velocity is changed to:

`bbV_(out)=[-v_x+2vp_x,v_y,v_z]`

and this causes a force, `F_p` to the piston of

`F_p=2m(v_x-vp_x)`

Gas Atom Hits the Metal Atoms at the Lower Boundary

The gas atom is reflected by the moving solid atoms which will have essentially infinite mass. Therefore gas atom does not transfer any of its energy to the solid atoms but, on average, the solid atom DOES transfer some of its energy to gas atom. Please note, when viewing the animation, oscillation of some metal atoms cause them to rise above the lower boundary of the cylinder. This is a normal situation in a heated solid: The upper boundary of the solid is not planar because of oscillation of its atoms about their anchor points. For this reason and the fact that the gas atom lower boundary always reflects the gas atoms, the only interaction of the gas with the solid atoms is when the solid atom rises above the gas lower boundary.

Gas Atom, `A_1`, Hits Another Gas Atom, `A_2`

This results in conservation of the sum of both the energy and the momentum of atoms `A_1` and `A_2`. First we define the vector distance between atoms `A_1` and `A_2`:

`[deltax,deltay,deltaz]=[A2.x-A1.x,A2.y-A1.y,A2.z-A1.z]`

and name the distance, `d`, just before collisions

`d=sqrt[deltax^2+deltay^2+deltaz^2)]`

then the following equations will compute the 3D velocity of the gas atom after the collision:

First we need the unit vector along line between centers.

`U=[u_x,u_y,u_z]=[(deltaz)/d,(deltay)/d,(deltaz)/d]`

We also need the vector difference in 3D velocity:

`deltaV=[deltav_x,deltav_y,deltav_z]=[A2.v_x-A1.v_x,A2.v_y-A1.v_y,A2.v_z-A1.v_z]`

For notation convenience let the atom masses be:

`A1.m=m_1`
`A2.m=m_2`

where `A.m` equals the mass of `A`. Now we need the reduced mass, `M`, of the atom pair

`M=(2m_1m_2)/(m_1+m_2)`

Then we take the dot procuct of unit vector with difference in 3D velocity and multiply by M

`MU*deltaV=M[u_x,u_y,u_z][[deltav_x],[deltav_y],[deltav_z]]`

Finally we have the 3D velocity CHANGE components of A1 and A2:

`deltav1_x=Mu_x(deltav_x)/m_1,deltav1_y=Mu_y(deltav_y)/m_1,deltav1_z=Mu_z(deltav_z)/m_1;`

and the the 3D velocity CHANGE components of A2

`deltav2_x=-Mu_x(deltav_x)/m_2,deltav2_y=-Mu_y(deltav_y)/m_2,deltav2_z=-Mu_z(deltav_z)/m_2;`

Finally velocity of A1 and A2 after collision:

`A1.vx+=deltavx_1,A1.vy+=deltavy_1,A1.vz+=deltavz_1`
`A2.vx+=deltavx_2,A2.vy+=deltavy_2,A2.vz+=deltavz_2`

Motion of the Solid Atoms in the Metal Plate

For this animation, the task of the solid atoms is just to provide energy to the gas atoms. In any solid the atoms (really the nucleii since they have almost all of the mass) move with respect to a fixed center. We will let this motion be sinusoidal with an amplitude `a` and a radian frequency `omega`. Now we need to describe the direction of this motion. To start, let the `z` axis be along the normal to the top gas boundary. The velocity can then be described as

`v_x=|bbv|cosphicostheta`
`v_y=|bbv|sinphicostheta`
`v_z=|bbv|sintheta`

where `|bbv|` is the magnitude of the velocity along the new axis, `theta` is the velocity axis angle with respect to the z axis and phi is the azimuthal angle with repect to the normal of the gas front boundary. For each solid atom we will randomly choose `theta` in the range `-pi/2 to pi/2` and `phi` in the range `0 to 2pi` The value of `|bbv|` will of course be time dependent

`|bbv(t)|=aomegacos(omegat+phi_0)`

where `phi_0` will be random in the range `0 to 2pi`. The values of the position of the solid atom are just the integrals with respect to time of their velocities: First we will take the integral of `|bbv(t)`

`int|bbv(t)|dt=intaomegacos(omegat+phi_0)dt=asin(omegat+phi_0)`

`x_s=asin(omegat+phi_0)cosphicostheta`
`y_s=asin(omegat+phi_0)sinphicostheta`
`x_s=asin(omegat+phi_0)sintheta`

Data while Running

Gas and cylinder with spring. Heating the gas causes its the gas atoms to speed up which increases their energy and pressure. The larger pressure impact on the piston accelerates it and the resulting motion compresses the attached spring. The Kelvin temperature increases proportionally with with the spring compression. Please note that you can pause heating and the piston will still move because the of its momentum and the gas pressure is still changing.

Charts of energy Versus time of various elements of the thermometer apparatus. Most important is the total energy of the apparatus. Note that the total energy includes the expended piston drag energy. The total energy changes rapidly during heating but it stops changing entirely when heating is stopped which proves that the system is conserving energy. When starting many of the elements have the similar values but their values can be read instead on the legend on the left.