Worksheet: Internal Energy

In this worksheet, we will practice describing the internal energy of an object in terms of the changes in the kinetic and potential energy of its particles.

Q1:

In the kinetic model of an ideal gas, the internal energy of a gas is modeled as consisting of only the kinetic energy of the particles of the gas. Which of the following statements most correctly describes why this approximation is effective in modeling some changes in real gases?

  • AThe potential energy of particles in a gas is all converted to kinetic energy by collisions between gas particles.
  • BGas particles travel at such high speeds that the forces between them act on them for negligible time intervals.
  • CThe internal energy of the gas is continuously being converted back and forth between kinetic energy and potential energy, so all the energy can be considered on average as kinetic energy.
  • DThe distances between particles in a gas are large enough for the average forces of attraction between the particles to be negligible.

Q2:

The specific latent heat of a substance is the energy required to change the phase of an object made of the substance that has a mass of one kilogram.

While an object is changing phase, does the average distance between its particles change?

  • ANo
  • BYes

While an object is changing phase, does the average kinetic energy of its particles change?

  • ANo
  • BYes

While an object is changing phase, does the average potential energy of its particles change?

  • AYes
  • BNo

While an object is changing phase, does its internal energy change?

  • ANo
  • BYes

While an object is changing phase, does its temperature change?

  • AYes
  • BNo

Q3:

Which of the following statements most correctly describes the internal energy 𝐸 of an object in terms of the net kinetic energy of its particles KEnet and the net potential energy of its particles PEnet?

  • A𝐸=+KEPEnetnet
  • B𝐸=−KEPEnetnet
  • C𝐸=KEnet
  • D𝐸=KEPEnetnet
  • E𝐸=PEKEnetnet

Q4:

The graph shows the ratio of potential energy to kinetic energy that makes up the internal energy of an object. Different temperatures 𝑇, 𝑇, 𝑇, and 𝑇 of the object correspond to vertical lines on the graph.

Which temperature corresponds to the melting point of the object?

  • A𝑇
  • B𝑇
  • C𝑇
  • D𝑇

Which temperature corresponds to the boiling point of the object?

  • A𝑇
  • B𝑇
  • C𝑇
  • D𝑇

In which region of the graph, I, II, III, IV, and V, is the object in the solid phase?

  • ARegion I
  • BRegion III
  • CRegion V
  • DRegion IV
  • ERegion II

In which region of the graph, I, II, III, IV, and V, is the object in the liquid phase?

  • ARegion IV
  • BRegion V
  • CRegion II
  • DRegion I
  • ERegion III

In which region of the graph, I, II, III, IV, and V, is the object in the gaseous phase?

  • ARegion I
  • BRegion III
  • CRegion II
  • DRegion V
  • ERegion IV

In which regions of the graph, I, II, III, IV, and V, is the object changing phase?

  • ARegions II and IV
  • BRegions II and III
  • CRegions III and V
  • DRegions III and IV
  • ERegions II, III, and IV

In which region of the graph, I, II, III, IV, and V, is the kinetic energy component of the object’s internal energy equal to the potential energy component?

  • ARegion II
  • BRegion I
  • CRegion III
  • DRegion V
  • ERegion IV

In region V of the graph, how does the ratio of the potential energy component of the object’s internal energy to the kinetic energy component change as the temperature of the object increases?

  • ADecreasing
  • BRemains constant
  • CIncreasing

Q5:

A solid object consists of four particles, as shown in the diagram. The particles oscillate around the mean positions that define the corners of a square. The object is heated and the mean separation of each particle from each of its nearest neighbors increases from 𝑑 to 𝑑.

Does the total potential energy of the object’s particles increase, decrease, or remain constant?

  • AIt increases.
  • BIt remains constant.
  • CIt decreases.

Does the total kinetic energy of the object’s particles increase, decrease, or remain constant?

  • AIt decreases.
  • BIt increases.
  • CIt remains constant.

Does the internal energy of the object increase, decrease, or remain constant?

  • AIt remains constant.
  • BIt increases.
  • CIt decreases.

Does the temperature of the object increase, decrease, or remain constant?

  • AIt increases.
  • BIt decreases.
  • CIt remains constant.

Does the object change phase?

  • ANo
  • BYes

Q6:

An object consists of two particles oscillating around mean positions that are separated by a distance 𝑑, as shown in the diagram. When the particles are at their greatest possible separation, 𝑑, they have velocities 𝑣. When the particles are at their smallest possible separation, 𝑑, they have velocities 𝑣. The speed of the particles while at their mean positions is 𝑠mean.

How does the magnitude of 𝑣 compare to 𝑠mean?

  • A𝑣>𝑠mean
  • B𝑣<𝑠mean
  • C𝑣=𝑠mean

How does the magnitude of 𝑣 compare to 𝑠mean?

  • A𝑣<𝑠mean
  • B𝑣>𝑠mean
  • C𝑣=𝑠mean

What is the total kinetic energy of the particles when their velocities are equal to 𝑣?

What is the total potential energy of the particles when their velocities are equal to 𝑣?

Q7:

The increase in potential energy of a particle as it increases its distance from another particle is shown in the graph. If the energy given to the particle, Δ𝐸, to increase its distance by Δ𝑑 is the same for each distance increase, how does the kinetic energy increase, ΔKE, of the particle over a distance Δ𝑑 compare with the kinetic energy increase that will occur for an additional Δ𝑑 increase in distance, ΔKE?

  • AΔ=ΔKEKE
  • BΔ>ΔKEKE
  • CΔ<ΔKEKE

Q8:

An object consists of two particles that are connected to each other, oscillating around points that are separated by a distance 𝑑, as shown in the diagram. The dashed circular lines show the maximum distance that the particles oscillate from their mean positions. The internal energy of the object is increased by Δ𝐸, and the distance between the particles increases by Δ𝑑 to 𝑑. The internal energy of the object is then further increased by Δ𝐸=Δ𝐸, and the distance between the particles further increases. The speeds of the particles when at the points separated by the distance 𝑑=𝑑+Δ𝑑 is equal to the speeds of the particles when their mean separation distance was 𝑑.

When the distance between the particles increases from 𝑑 to 𝑑, is the average attractive force that the particles exert on each other equal to, less than, or greater than the average attractive force that the particles exert on each other when the distance between the particles increases from 𝑑 to 𝑑?

  • ALess than
  • BEqual to
  • CGreater than

When the distance between the particles increases from 𝑑 to 𝑑, does the temperature of the object increase?

  • AYes
  • BNo

When the distance between the particles increases from 𝑑 to 𝑑, does the temperature of the object increase?

  • ANo
  • BYes

Q9:

Liquid water has a density of 997 kg/m3, whereas water vapor has a density of 0.804 kg/m3. 1 kilogram of water contains 3.346×10HO2 molecules. The water vapor is heated, expanding its volume to 50 m3. The force exerted by particles on each other is proportional to 1𝑑, where 𝑑 is the distance between the particles.

How many times greater is the average distance between adjacent HO2 molecules in gas phase than in liquid phase before the gas is heated? Answer to three significant figures.

How many times greater is the average distance between adjacent HO2 molecules in gas phase than in liquid phase after the gas is heated? Answer to three significant figures.

What is the ratio of the average attractive force between adjacent particles in the liquid phase to the average attractive force between adjacent particles in the gas phase before the gas is heated? Answer to three significant figures.

What is the ratio of the average attractive force between adjacent particles in the liquid phase to the average attractive force between adjacent particles in the gas phase after the gas is heated? Answer to three significant figures.

Q10:

A solid object contains many particles. Two of these particles oscillate around points that are separated by a distance 𝑑, as shown in the diagram, moving a mean speed 𝑠mean. The mean particle speed is represented in the diagrams by the radii, 𝑟, of the dashed circles that correspond to the greatest distance that the particles move from the centers of their oscillations. The object containing the particles is heated and the distance between the particles increases to 𝑑, while the value of 𝑠mean increases. The object is then heated further. After this additional heating, the particles are separated by a maximum distance 𝑑, where 𝑑>2𝑑. The particles attract each other and the distance between them is reduced to 𝑑.

When the distance between the particles increases from 𝑑 to 𝑑, does the total kinetic energy of the particles increase?

  • ANo
  • BYes

When the distance between the particles increases from 𝑑 to 𝑑, does the total potential energy of the particles increase?

  • AYes
  • BNo

When the distance between the particles increases from 𝑑 to 𝑑, does the total kinetic energy of the particles increase?

  • AYes
  • BNo

When the distance between the particles increases from 𝑑 to 𝑑, does the total potential energy of the particles increase?

  • AYes
  • BNo

When the distance between the particles is 𝑑, how does their speed, 𝑠, compare to the maximum speed, 𝑠max, that the particles had when the mean distance between them was 𝑑?

  • A𝑠<𝑠max
  • B𝑠=𝑠max
  • C𝑠>𝑠max

When the distance between the particles is 𝑑, is their total potential energy greater than, less than, or equal to zero?

  • AEqual to zero
  • BLess than zero
  • CGreater than zero

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