Mass defect & binding energy

A hot air balloon consists of a fabric envelope, heaters and a basket, as shown.

When the balloon is set up, the envelope is partly filled with air at 20°C. The air is then heated to 120°C and expands to fill the envelope and becomes less dense.

The air pressure inside the envelope is always equal to the air pressure outside the envelope because the envelope is open at the bottom.

The balloon takes off when the upthrust is more than the total weight of the balloon, the air in the envelope and the passengers.

Deduce whether the balloon can take off.

volume of air at 120°C in inflated envelope = 2800 m³ density of air at 20°C = 1.2 kg m^–3 mass of balloon = 380 kg mass of passengers = 340 kg upthrust when the envelope is full = 33 000 N

State one assumption of the kinetic theory of gases.

Derive an equation to show that, for a gas at temperature T, the mean kinetic energy of the molecules = kT

Calculate the root-mean-square speed of nitrogen molecules at a temperature of 120°C.

mass of nitrogen molecule = 28 u

A fusion research centre was opened in Rotherham in 2021. The centre has a device which tests materials in the extreme conditions found inside a fusion reactor.

Describe the extreme conditions inside a fusion reactor.

The graph shows how the binding energy per nucleon varies with nucleon number for a range of nuclides.

Explain why the fusion of nuclei can produce large amounts of energy. Your answer should refer to information from the graph.

In a particular radioactive decay, there is a mass decrease equivalent to 0.05 u.

Which of the following expressions gives the energy released in MeV?

The photograph shows a vase made of uranium glass. Uranium glass is radioactive.

Uranium glass usually contains a maximum of 2% uranium. Uranium glass made in the early part of the 20th century can contain up to 25% uranium.

A student carried out an investigation to determine the percentage of uranium in the glass.

The student measured the count rate by placing a Geiger Muller (GM) tube against the vase at a single position. This value was used to calculate the decay rate for the whole vase.

Show that the decay constant for uranium is about 5 10⁻¹⁸s⁻¹

half-life of uranium = 1.41 10¹⁷s

Calculate the percentage of uranium, by mass, in the glass.

area of GM tube window = 6.36 10⁻⁵m² surface area of vase = 0.0177 m² background count rate = 525 counts in 10 minutes count rate when GM tube next to vase = 3623 counts in 5 minutes mass of vase = 149 g mass of uranium atom = 238 u

The uranium decays by emitting alpha particles.

Criticise the method used to determine the percentage of uranium in the vase.

A uranium nucleus decays to thorium by emission of an alpha particle.

It can be assumed that all the energy of the decay is transferred to kinetic energy of the alpha particle.

Calculate the speed of the emitted alpha particle.

mass of uranium nucleus = 238.0003 u mass of thorium nucleus = 233.9942 u mass of alpha particle = 4.0015 u

At the Culham Centre for Fusion Energy (CCFE) experiments are carried out to investigate nuclear fusion and the properties of plasmas. A plasma consists of ionised gas, containing positive ions and electrons.

In a fusion experiment at CCFE, ions of two isotopes of hydrogen fuse to produce helium ions and fast-moving neutrons.

Show that a single fusion reaction releases about 3 10⁻¹²J of energy.

mass of₁²H = 2.013553 u mass of₁³H = 3.015501 u mass of⁴₂He = 4.001506 u mass of₀¹n = 1.008665 u

Fusion occurs naturally in the core of stars.

Explain why very high densities of matter and very high temperatures are needed to bring about and maintain nuclear fusion in stars.

In a plasma experiment 5.0 mg of deuterium, an isotope of hydrogen, occupies a volume of 98 m³. The temperature of deuterium is raised to 1.3 10⁸K. In this experiment, the deuterium behaves as an ideal gas.

Calculate the pressure due to the deuterium ions.

mass of deuterium ion = 3.3 10⁻²⁷kg

Calculate the root mean square speed of the deuterium ions at this temperature.

The temperature of the plasma is monitored using the Doppler effect. Light from a laser is directed into the plasma and the wavelength of the light reflected is measured.

The Doppler shift observed when light is reflected by a deuterium ion is twice the Doppler shift that would be observed for a source of light moving at the same speed as the deuterium ion.

Calculate the maximum wavelength of light that would be detected after reflection from a deuterium ion moving at 1.5 × 10⁶m s⁻¹.

wavelength of laser light = 1064 nm

At the Culham Centre for Fusion Energy (CCFE) experiments are carried out to investigate nuclear fusion and the properties of plasmas. A plasma consists of ionised gas, containing positive ions and electrons.

Fusion occurs naturally in the core of stars.

Explain why very high densities of matter and very high temperatures are needed to bring about and maintain nuclear fusion in stars.

In a plasma experiment 5.0 mg of deuterium, an isotope of hydrogen, occupies a volume of 98 m³. The temperature of deuterium is raised to 1.3 10⁸K. In this experiment, the deuterium behaves as an ideal gas.

Calculate the pressure due to the deuterium ions.

mass of deuterium ion = 3.3 10⁻²⁷kg

Calculate the root mean square speed of the deuterium ions at this temperature.

The temperature of the plasma is monitored using the Doppler effect. Light from a laser is directed into the plasma and the wavelength of the light reflected is measured.

The Doppler shift observed when light is reflected by a deuterium ion is twice the Doppler shift that would be observed for a source of light moving at the same speed as the deuterium ion.

Calculate the maximum wavelength of light that would be detected after reflection from a deuterium ion moving at 1.5 × 10⁶m s⁻¹.

wavelength of laser light = 1064 nm

*4 Radium is a radioactive element. The most common isotope of radium has a half-life of almost two thousand years. A sample of radium can remain at a higher temperature than its surroundings for a long period of time.

Explain how a sample of radium is able to release significant amounts of energy over a long period of time.

*4 Radium is a radioactive element. The most common isotope of radium has a half-life of almost two thousand years. A sample of radium can remain at a higher temperature than its surroundings for a long period of time.

Explain how a sample of radium is able to release significant amounts of energy over a long period of time.

Nuclear fusion involves small nuclei joining to make larger nuclei. Nuclear fission involves large nuclei splitting to become smaller nuclei. Both of these processes release energy.

The graph shows how the binding energy per nucleon varies with nucleon number for a range of isotopes.

Use the binding energy per nucleon curve to explain how fusion and fission both release energy.

Explain the conditions required to bring about and maintain nuclear fusion.

Mass spectrometry is a technique used to separate ions based on their charge to mass ratio.

The atoms in a sample are ionised and then accelerated and formed into a fine beam. This beam is passed into a region of uniform magnetic field and the ions are deflected by different amounts according to their mass.

Analysis of mass spectrometer data shows that chlorine exists in nature as two isotopes, chlorine-35 and chlorine-37.

State what is meant by isotopes.

In a mass spectrometer, chlorine-35 ions are accelerated through a potential difference of 8.50 kV to produce an ion beam.

Show that the speed of singly ionised chlorine-35 atoms is about 2.2 × 10⁵ m s⁻¹.

mass of an ion of chlorine-35 = 34.97 u

In most mass spectrometers the ions are passed through a velocity selector, after being accelerated, to produce a beam of ions of a particular velocity. The velocity selector consists of a pair of parallel plates, across which a potential difference (p.d.) is applied to create an electric field.

In one mass spectrometer the plates are 2.5 cm apart and a p.d. of 135 V is applied.

A magnetic field is also applied to produce a force on the ions in the opposite direction to the force from the electric field. For one particular speed the ions travel in a straight line and emerge from the selector.

Add to the diagram to indicate the directions of the electric field and the magnetic field.

The magnetic flux density applied to the velocity selector is 24.5 mT.

Deduce whether this magnetic flux density is suitable to produce a beam of chlorine-35 ions of speed 2.2 × 10⁵ m s⁻¹.

After passing through the velocity selector the ion beam enters a region of uniform magnetic flux density 0.35 T with the ions travelling at right angles to the field direction.

Explain why the ions travel in a circular path.

Calculate the radius of the circular path.

The diagram shows the path of the chlorine-35 ions in the field. Chlorine-37 ions enter the magnetic field with the same velocity.

1. Add another line to the diagram to show the path of these chlorine-37 ions.

2. Explain any differences in the paths.