In this question, we're presented with a diagram that shows us the change in mass of an isotope as a function of time.

To find the half-life, we first need to find the point on the y-axis where the mass is half of its starting value. Since  of the isotope is present initially, we'll have to direct our attention to the point on the y-axis that shows  of the isotope.

Next, we'll need to see where this value on the y-axis matches up with the corresponding value on the x-axis. From the graph, we can see that the  point of the y-axis matches up approximately with the  mark on the x-axis. Thus, we can conclude that the time it takes for half of the isotope to decay is equal to , which equates to a half-life of .

For this question, we need to determine which graph correctly represents the decay of a radioactive isotope over time. First, recall that all radioactive decay processes are first-order reactions. Thus, they take the general form:

Where  represents the rate constant for the reaction. Thus, at any given time during the decay of the unstable isotope, the rate is dependent on the concentration of the isotope present. Therefore, as the decay proceeds, there will be less and less of the isotope present as time goes on. This, in turn, means that the rate will also drop proportionately as the concentration of isotope goes down. This would result in a graph in which the decay process begins rapidly, but then begins to decline and level off as time goes on.

Another way of looking at this is to note that on a graph of isotope concentration vs. time, the slope of the line tangent to any point on the graph represents the rate of decay at that point. Thus, the correct graph will have a steep slope initially, but as time goes on (as the value of x on the graph becomes larger), the slope will decrease until it levels off.

If radium experiences alpha decay, it will lose four total nucleons and two protons. Thus, the mass number will decrease by 4 and the atomic number will decrease by 2.

Using

Solving for

Plugging in values:

Use the equation for radioactive decay:

Solving for :

Plugging in values:

Use the equation for radioactive decay:

Solving for :

Plugging in values:

Use the following relationship:

Plugging in values:

Use the equation for radioactive decay:

Solving for :

Plugging in values:

Again using the first equation:

Plugging in values:

Use the equation for radioactive decay:

Solving for :

Plugging in values:

Use the following two relationships:

and

Combining equations:

Solving for :

Plugging in values:

To find the approximate age of the sample, we need to consider the half-life of the starting material, substance A. We also need to consider the relative amounts of substance A and substance B.

Since we are told that the sample we are looking at contains  substance A and  substance B, we know that the sample must have started out with  of substance A. Furthermore, in order to go from  to , we know that a total of two half-lives must have passed. And since we know that one half-life is equal to , we can conclude that a total of  must have passed.

In this question, we're shown a reaction in which platinum is decaying into some unknown element. We're asked to determine how many neutrons this unknown element will have.

In the reaction shown in the question stem, we can see that platinum has  protons, which is its atomic number. The other number shown, , is the mass number. The mass number represents the total number of neutrons and protons. Since we know the mass number and the atomic number, we can determine the number of neutrons in platinum.

Thus, there are  neutrons in platinum. To determine the number of neutrons in the product, however, we need to look at which type of decay we're dealing with. Since an electron is being produced in the reaction shown, we know that this must be beta-decay. In this type of decay, a neutron within the nucleus of the starting atom (platinum) is broken down into a proton and an electron, which is then emitted. Consequently, the product element should have one less neutron that the starting element, platinum. Thus, we can conclude that there will be  neutrons in the new element.