10: Modelling decay Flashcards
What makes nucleus unstable?
The nucleus becomes unstable when:
- Too many neutrons
- Not enough neutrons
- Too many nucleons in total
- Too much energy
What happens to unstable nuclei?
Unstable nuclei break down by releasing energy and/or particles, until they reach a stable form – this process is called radioactive decay
Is radioactive decay a probability or random process? Why?
Radioactive decay is a random process - you can’t tell when any one nucleus will decay, or which nucleus in a sample will be the next to decay.
in radioactive decay a particle has a probability of decaying but this event is absolutely random you can’t tell when anyone nucleus will decay, or which nucleus in a sample will be the next to decay.
Describe how radioactive decay can be modelled
Although you can’t predict when a nucleus will decay, you can still make predictions about the behaviour of a source using a model - exponential decay is based on a very large number of undecayed nuclei.
If you take a large enough sample of unstable nuclei, the overall behaviour shows a pattern. This means you can predict how many nuclei will decay in a given time.
Define activity and units
The activity of a source is the number of nuclei decaying per second (easier to use activity)
Measured in becquerels Bq 1Bq = 1 decay per second
What is the activity of a sample proportional to? Why?
The activity of a sample is proportional to size of the sample available (number of undecayed nuclei)
The size – as nuclei decay, the sample size gets smaller, so the activity falls
what is half life
the time taken for the number of undecayed nuclei remaining to halve, or for the activity or count rate to halve
how to find final activity having been given original activity and number of half lives done
final activity = original activity / 2^number of done half lives
what is count rate
the number of particles emitted from the sample that the system detects per second
Briefly describe the equipment used in the investigation of count rate
A Geiger counter attached to the Geiger-Müller tube which is on a clamp and stand. A protactinium generator in front of the tube.
how will count rate differ from activity and what is their relationship
count rate will usually be smaller than the activity because not all emitted nuclei will be detected
however count rate is proportional to activity, if activity halves, count rate halves.
What does the decay constant measure?
the fixed probability of a given nucleus decaying in a certain time interval. The bigger the value of the decay constant, the more likely a decay is, so the faster the rate of decay.
how to find half life from a graphd
Find the value of undecayed nuclei when t=0 Go to half the original value of N Work out the corresponding time (You can check this by finding the quarter of N)
how to find the undecayed nuclei N remaining after L half lives
N = 2^L number of undecayed nuclei remaining = 2^number of half lives
equation to find activity
activity = λN
activity = fixed prob * nucleus decaying * number of undecayed nuclei
how to determine the HL of protactinium
before you measure activity of source, measure background count sources to remove systematic error
use apparatus to record counts over 10 seconds for 3 minutes
find count rate by dividing counts in each period by 10 (to get count rate per second), now find correct count rate by subtracting this count rate from the background count rate
draw graph of count rate (y) against time (x) to find half life
iterative equation for number of nuclei decaying in a time interval what does the negative sign show what type of equation is this
ΔN / Δt = -λN the negative sign shows the number of nuclei decreases after every decay
this is a differential equation
what can we say about the relationship between the number of nuclei remaining and activity using the differential equation
the number of nuclei remaining is proportional to the activity ΔN / Δt ∝ -N
what is a disadvantageous assumption made with iterative models for radioactive decay, and what is reality
how can you improve your iterative model
the line between each calculated value of nuclei remaining at a certain time is straight so the model assumes that the activity (and so number of nuclei) is constant during each time interval.
however really the rate of decay falls continuously as the number of undecayed nuclei remaining falls.
you can improve the model by reducing the time interval between calculations
how to calculate half life if given decay constant
T1/2 = ln2 / decay constant
what is the rearranged equation of the differential equation to find number of undecayed nuclei remaining and activity at a point
ΔN / Δt = -λN goes to
N = N0 e^-λt
N = number of radioactive nuclei
e = exponential constant
λ = decay constant
t = time (s)
A = A0 e^-λt
A = activity
e = exponential constant
λ = decay constant
t = time (s)
graphical form of A = A0 e^-λt what does each part represent
lnA = -λt + lnA0
y =mx + c
lnA = y
lnA0 = c
m = -λ (decay constant is positive, the gradient gives you negative decay constant)
x = t
What is a capacitor? What is it made of?
Capacitors also store energy
Capacitors are made of two metal plates on which charge is built up and stored, separated by an air gap so that charge doesn’t flow between them.
electrons flow across the circuit from one plate to another, building up opposite charges
Describe how capacitors store energy (using the circuit with battery, capacitor and bulb)
When the switch is flicked (so that the battery and capacitor are connected), charge builds up on the plates of the capacitor.
Energy provided by the battery is stored, in the capacitor, in the form of electrical potential energy
When the switch is flipped the other way, the energy stored on the plates fully discharge through the bulb, converting electrical potential energy into light and heat
Describe what work is done on a capacitor charging
Work is done removing charge from one plate and depositing charge onto the other one. The energy for this must come from the energy stored in the battery
Define capacitance
The amount of charge stored per volt
capacitance equation units
C = Q / V capacitance C is in Farads, F which is C V^-1
conversion of microfarads, nano-farads and picofarads to farads
microfarads = uF = 1*10^-6 farads
nanofarads = nF = 1*10^-9 farads
picofarads = pF = 1*10^-12 farads
how do you investigate discharge of a capacitor
cap = 2200 uF
resistor = 10000 ohms
draw circuit as shown, you can use this to observe how pd across capacitor changes with time as it discharges
capacitor is charged at A
when at B it will begin discharging, as soon as this begins, start timing
for cap = 2200 uF and resistor = 10000 ohms, take readings every 5 seconds
for smaller values of either, take more frequent readings (could use data logger)
analyse data to draw graphs showing variation of charge, pd, current with time
What does the graph of potential difference against charge look like? What is the area?
Straight line through the origin, area = energy
Describe the graph of current/voltage/charge against time for a discharging capacitor.
All the same graph shape All start of high when t=0. All done exponentially decrease as time increases
Describe the current of the discharging capacitor. Investigation of what happens when you charge/discharge a capacitor
The current flows in the opposite direction to the charging current. As the potential difference decreases, the current decreases as well
Capacitor charging: Describe the graph of current against time
Current starts high when t=0 and exponentially decrease to 0 as time increases due to charge building up on the oppositely charged plates which increases electrostatic repulsion making it harder for more electrons to be deposited, and when pd across capacitor is finally equal to the pd across the battery supply, current is 0
Capacitor charging: Describe the graph of voltage against time
Same as charge against time Voltage starts at zero, initially increases rapidly as time increases, then voltage increases slowly as time increases
Capacitor charging: Describe the graph of charge against time
Same as voltage against time Charge starts at zero, initially increases rapidly as time increases, then charge increases slowly as time increases
investigation for charging capacitor
make sure capacitor is fully discharged by connecting a lead across terminals of capacitor before charging
it will begin charging when switch is closed
record pd across it in equal time intervals, using data logger for short time intervals
differential equation for rate of discharge
ΔQ / Δt = -Q / RC
resistance and capacitance are constant so rate of change of charge is proportional to the charge remaining
diff equation to find charge, pd, current for discharging
diff equation to find charge, pd, current for charging
What does the time it take to charge or discharge a capacitor depend on?
The capacitance of the capacitor. This affects the amount of charge that can be transferred for given potential difference
The resistance of the circuit. This affects the current in a circuit
What is the time constant? Capacitors
The time taken for the charge, potential difference or current of a discharging capacitor to fall to 37% of its initial value
The time taken for the charge or potential difference of a charging capacitor to rise to 63% of its value when fully charged
What is the relationship between the resistance in series with the capacitor and the time it takes a capacitor to charge or discharge?
The larger the resistance in series with the capacitor, the longer the capacitor takes to charge or discharge
Why do you iterative methods only give you approximate answers?
What makes an iterative of method more accurate?
What is the disadvantage of using a smaller time interval, when doing iterative methods?
They assume that dx/dt doesn’t change over the time interval you’re using. In fact, dx/dt is changing all the time
Use a smaller value for the time interval
You need to do more iterations
