Question

which of the following correctly converts 15.0 °c to kelvin using correct rounding and significant figures? 288 k 290 k 288.15 k 288.2 k your answer is incorrect. try again

Explanation:

Step1: Recall the formula for converting Celsius to Kelvin

The formula to convert temperature from Celsius (\(^{\circ}\text{C}\)) to Kelvin (K) is \(T(K)=T(^{\circ}\text{C}) + 273.15\).

Step2: Substitute the given Celsius value into the formula

We are given \(T(^{\circ}\text{C}) = 15.0^{\circ}\text{C}\). Substituting this into the formula:
\(T(K)=15.0 + 273.15\)
\(T(K)=288.15\space K\)

But we also need to consider significant figures and rounding. The given value \(15.0^{\circ}\text{C}\) has three significant figures. When we add \(15.0\) (three significant figures) and \(273.15\) (five significant figures), the result should be rounded to three significant figures? Wait, no, when adding, the number of decimal places matters. \(15.0\) has one decimal place, \(273.15\) has two decimal places. The result should be rounded to one decimal place? Wait, no, let's re - evaluate.

Wait, the formula is \(K = ^{\circ}\text{C}+273.15\). For \(15.0^{\circ}\text{C}\):

\(15.0 + 273.15=288.15\space K\). But if we consider significant figures, \(15.0\) has three significant figures. The value \(288.15\) can be rounded to \(288\space K\) (if we take three significant figures) or \(288.2\space K\) (if we take one decimal place) or \(290\space K\) (if we round to two significant figures, but \(15.0\) has three). Wait, maybe the question is about the exact conversion first.

Wait, the exact conversion of \(15^{\circ}\text{C}\) to Kelvin is \(15 + 273.15 = 288.15\space K\). But if we consider the given value \(15.0^{\circ}\text{C}\) (three significant figures), when we add \(15.0\) and \(273.15\), the sum is \(288.15\). Now, let's check the options:

- Option 288 K: If we round \(288.15\) to three significant figures, it is \(288\space K\) (since the fourth digit is 1, which is less than 5, we round down).
- Option 290 K: Rounding \(288.15\) to two significant figures gives \(290\space K\) (but \(15.0\) has three significant figures, so this is incorrect).
- Option 288.15 K: This is the exact value from the formula.
- Option 288.2 K: Rounding \(288.15\) to one decimal place gives \(288.2\space K\).

But the most accurate conversion without improper rounding (considering the formula) is \(288.15\space K\), but if we consider significant figures based on \(15.0\) (three significant figures), \(288\space K\) is also a candidate. Wait, there is a mistake in my earlier thought. The formula \(K = ^{\circ}\text{C}+273.15\) is an exact conversion factor (the 273.15 is exact in the definition of the Kelvin scale relative to Celsius). So when converting \(15.0^{\circ}\text{C}\) (which has three significant figures), the result should have three significant figures. \(288.15\) rounded to three significant figures is \(288\space K\) (because the digit after the third significant figure is 1, which is less than 5).

Wait, let's recalculate:

\(15.0+273.15 = 288.15\)

Now, for significant figures:

- The number \(15.0\) has three significant figures.
- When adding, the result should be reported to the least number of decimal places. \(15.0\) has one decimal place, \(273.15\) has two decimal places. So the result should be reported to one decimal place. \(288.15\) rounded to one decimal place is \(288.2\space K\).

But now we have a conflict. Let's check the options again. The options are 288 K, 290 K, 288.15 K, 288.2 K.

The exact conversion of \(15^{\circ}\text{C}\) to Kelvin is \(15 + 273.15=288.15\space K\). If the question is about the exact conversion (without considering significant figures for rounding down), then \(288.15\space K\) is the exact value. But if we consider that \(15.0^{\circ}\text{C}\) (three significant figures) and we want to present the answer with three significant figures, \(288\space K\) (since \(288.15\approx288\) when rounded to three significant figures).

Wait, maybe the question is not about significant figures in the sense of rounding down for three significant figures, but just the conversion formula. Let's go back to the formula: \(K = ^{\circ}\text{C}+273.15\). For \(15.0^{\circ}\text{C}\), \(K = 15.0+273.15 = 288.15\space K\). But if we round to a reasonable number, if we take the given \(15.0\) (which has three significant figures), the sum \(288.15\) can be written as \(288\space K\) (three significant figures) or \(288.2\space K\) (one decimal place) or \(288.15\space K\) (exact).

Now, let's check the options:

- 288 K: \(288.15\) rounded to three significant figures is \(288\) (since the fourth digit is 1 < 5).
- 290 K: This is a wrong rounding (rounding \(288.15\) to two significant figures gives \(290\), but \(15.0\) has three, so this is incorrect).
- 288.15 K: Exact value from the formula.
- 288.2 K: Rounded to one decimal place.

But the question says "using correct rounding and significant figures". The value \(15.0^{\circ}\text{C}\) has three significant figures. When we add \(15.0\) (three significant figures, one decimal place) and \(273.15\) (five significant figures, two decimal places), the result should be reported to one decimal place (because the least number of decimal places in the terms being added is one). So \(288.15\) rounded to one decimal place is \(288.2\space K\). Wait, \(15.0\) has one decimal place, \(273.15\) has two. The rule for addition is that the result has the same number of decimal places as the term with the least number of decimal places. So \(15.0\) (1 decimal place) + \(273.15\) (2 decimal places) = \(288.15\), which should be rounded to 1 decimal place: \(288.2\space K\).

Wait, now I'm confused. Let's start over.

The formula for converting Celsius to Kelvin is \(K = ^{\circ}\text{C}+273.15\).

Given \(T(^{\circ}\text{C})=15.0^{\circ}\text{C}\)

\(T(K)=15.0 + 273.15=288.15\space K\)

Now, significant figures: \(15.0\) has three significant figures. The number \(288.15\) - let's see the significant figures. The first non - zero digit is 2, then 8, 8, 1, 5. So if we take three significant figures, it's \(288\space K\) (since the fourth digit is 1, which is less than 5, we round down). If we take four significant figures, it's \(288.2\space K\) (rounding \(288.15\) to four significant figures: the fourth digit is 1, the next digit is 5, so we round up the fourth digit: \(288.2\)).

But the options are:

- 288 K
- 290 K
- 288.15 K
- 288.2 K

Now, let's check the exact conversion first. The exact conversion of \(15^{\circ}\text{C}\) (without the decimal) is \(15 + 273.15 = 288.15\space K\). But the given value is \(15.0^{\circ}\text{C}\), which is more precise.

If we consider that the question might be expecting the exact conversion (without considering significant figures for rounding down to three significant figures), then \(288.15\space K\) is the exact value. But if we consider significant figures, \(15.0\) has three significant figures, so the answer should have three significant figures. \(288.15\) rounded to three significant figures is \(288\space K\).

Wait, maybe the question is not about significant figures in the sense of reducing the number of significant figures, but just the conversion. Let's check the options again.

The correct conversion of \(15.0^{\circ}\text{C}\) to Kelvin using the formula \(K = ^{\circ}\text{C}+273.15\) is \(15.0 + 273.15=288.15\space K\). But if we round to a whole number, \(288\space K\) (since \(288.15\approx288\)).

But there is a mistake in my earlier analysis. Let's check the options:

- 288 K: \(15 + 273=288\) (if we use \(273\) instead of \(273.15\) for conversion, which is an approximate conversion).
- 290 K: Incorrect, as \(15 + 273 = 288\), not 290.
- 288.15 K: Exact conversion using \(273.15\).
- 288.2 K: Rounded to one decimal place.

The formula for Kelvin from Celsius is \(K = ^{\circ}\text{C}+273.15\) (exact definition). So for \(15.0^{\circ}\text{C}\), \(K = 15.0+273.15 = 288.15\space K\). But if we use the approximate formula \(K = ^{\circ}\text{C}+273\), then \(K = 15 + 273=288\space K\).

Now, the question says "using correct rounding and significant figures". The value \(15.0^{\circ}\text{C}\) has three significant figures. The conversion factor \(273.15\) is a defined constant (exact in the context of this conversion). So when we add \(15.0\) (three significant figures) and \(273.15\) (exact), the result should be reported with three significant figures. \(288.15\) rounded to three significant figures is \(288\space K\) (because the digit in the thousandths place is 1, which is less than 5, so we round down the hundredths place? Wait, no, significant figures are counted from the first non - zero digit. So \(288.15\) has five significant figures. To three significant figures, it's \(288\) (the first three digits are 2, 8, 8; the next digit is 1, so we keep the third digit as 8).

So the correct answer should be 288 K? Wait, no, let's check with an example. If the temperature is \(0^{\circ}\text{C}\), converting to Kelvin gives \(273.15\space K\), which is \(273\space K\) when rounded to three significant figures.

So, putting it all together, the conversion of \(15.0^{\circ}\text{C}\) to Kelvin is \(15.0 + 273.15 = 288.15\space K\). If we round to three significant figures, it is \(288\space K\).

Answer:

288 K