Think about what happened to the scale. For $\ce{^{16}O}$, it decreased from $16.0000\,\mathrm{amu}$ to $15.9994\,\mathrm{amu}$. So the scale was stretched out (since 0 needs to stay constant). That would mean that $\ce{^{12}C}$ would need to decrease from some number greater than $12.0000\,\mathrm{amu}$. With $0$ being constant, there is no addition factor (like when converting from Fahrenheit to Centigrade); it is just a ratio from one scale to the other.
$$\frac {\text{unknown}}{16.0000\,\mathrm{amu}}=\frac {12.0000\,\mathrm{amu}}{15.9994\,\mathrm{amu}}$$
Think about what happened to the scale. For $\ce{^{16}O}$, it decreased from $16.0000\,\mathrm{amu}$ to $15.9994\,\mathrm{amu}$. So the scale was stretched out (since 0 needs to stay constant). That would mean that $\ce{^{12}C}$ would need to decrease from some number greater than $12.0000\,\mathrm{amu}$. With $0$ being constant, there is no addition factor (like when converting from Fahrenheit to Centigrade); it is just a ratio from one scale to the other.$$\frac {\text{unknown}}{16.0000\,\mathrm{amu}}=\frac {12.0000\,\mathrm{amu}}{15.9994\,\mathrm{amu}}$$
Here is another equivalent way of thinking about the problem. The mass of oxygen on the old scale is 16 amu. The mass of oxygen on the new scale is 15.994 amu. When you divide them:
Basically, taking a ratio of any value in the old scale to the equivalent value in the new scale (or vice versa) gives you a conversion factor. So for every new amu, there is 1.0000375 old amus. The question is asking you to convert 12 new amus to old amus. Just by looking at the units, it should be obvious what to do:
If you explicitly label old and new, it's easy to see that the new units cancel out.
Incidentally, if you keep track of your significant figures, you can see why this redefinition of the amu didn't cause too many problems for chemists—even to five sig figs, the change is negligible.
Here is another equivalent way of thinking about the problem. The mass of oxygen on the old scale is 16 amu. The mass of oxygen on the new scale is 15.994 amu. When you divide them:
Basically, taking a ratio of any value in the old scale to the equivalent value in the new scale (or vice versa) gives you a conversion factor. So for every new amu, there is 1.0000375 old amus. The question is asking you to convert 12 new amus to old amus. Just by looking at the units, it should be obvious what to do:
If you explicitly label old and new, it's easy to see that the new units cancel out.
Incidentally, if you keep track of your significant figures, you can see why this redefinition of the amu didn't cause too many problems for chemists—even to five sig figs, the change is negligible.
Yeah I know, but Ive seen amu used as the unified atomic mass unit in general chemistry textbooks a lot so I used new and old for this example to make it clearer. Personally, I prefer Da to u, if for no other reason than "daltons" is easier to say than "unified atomic mass units".More
Think about what happened to the scale. For $\ce{^{16}O}$, it decreased from $16.0000\,\mathrm{amu}$ to $15.9994\,\mathrm{amu}$. So the scale was stretched out (since 0 needs to stay constant). That would mean that $\ce{^{12}C}$ would need to decrease from some number greater than $12.0000\,\mathrm{amu}$. With $0$ being constant, there is no addition factor (like when converting from Fahrenheit to Centigrade); it is just a ratio from one scale to the other. $$\frac {\text{unknown}}{16.0000\,\mathrm{amu}}=\frac {12.0000\,\mathrm{amu}}{15.9994\,\mathrm{amu}}$$
Think about what happened to the scale. For $\ce{^{16}O}$, it decreased from $16.0000\,\mathrm{amu}$ to $15.9994\,\mathrm{amu}$. So the scale was stretched out (since 0 needs to stay constant). That would mean that $\ce{^{12}C}$ would need to decrease from some number greater than $12.0000\,\mathrm{amu}$. With $0$ being constant, there is no addition factor (like when converting from Fahrenheit to Centigrade); it is just a ratio from one scale to the other.$$\frac {\text{unknown}}{16.0000\,\mathrm{amu}}=\frac {12.0000\,\mathrm{amu}}{15.9994\,\mathrm{amu}}$$
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Here is another equivalent way of thinking about the problem. The mass of oxygen on the old scale is 16 amu. The mass of oxygen on the new scale is 15.994 amu. When you divide them:
$$\frac{16\ \textrm{old}}{15.9994\ \textrm{new}} = 1.0000375 \frac{\textrm{old}}{\textrm{new}}$$
Basically, taking a ratio of any value in the old scale to the equivalent value in the new scale (or vice versa) gives you a conversion factor. So for every new amu, there is 1.0000375 old amus. The question is asking you to convert 12 new amus to old amus. Just by looking at the units, it should be obvious what to do:
$$\frac{16\ \textrm{old}}{15.9994\ \textrm{new}} * 12\ \textrm{new}=12.00045\ \textrm{old}$$
If you explicitly label old and new, it's easy to see that the new units cancel out.
Incidentally, if you keep track of your significant figures, you can see why this redefinition of the amu didn't cause too many problems for chemists—even to five sig figs, the change is negligible.
Here is another equivalent way of thinking about the problem. The mass of oxygen on the old scale is 16 amu. The mass of oxygen on the new scale is 15.994 amu. When you divide them:
$$\frac{16\ \textrm{old}}{15.9994\ \textrm{new}} = 1.0000375 \frac{\textrm{old}}{\textrm{new}}$$
Basically, taking a ratio of any value in the old scale to the equivalent value in the new scale (or vice versa) gives you a conversion factor. So for every new amu, there is 1.0000375 old amus. The question is asking you to convert 12 new amus to old amus. Just by looking at the units, it should be obvious what to do:
$$\frac{16\ \textrm{old}}{15.9994\ \textrm{new}} * 12\ \textrm{new}=12.00045\ \textrm{old}$$
If you explicitly label old and new, it's easy to see that the new units cancel out.
Incidentally, if you keep track of your significant figures, you can see why this redefinition of the amu didn't cause too many problems for chemists—even to five sig figs, the change is negligible.
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