# Measurement of Physical Quantities Using Appropriate Instruments

MEASUREMENT OF PHYSICAL QUANTITIES USING    APPROPRIATE INSTRUMENTS

Mass is a measure of how heavy something is i.e., it is the amount of matter present in it. It can  Be determined in the laboratory by using analytical or electrical balances as shown in Figs.2.1,  2.2 and 2.3.

• Mass is measured in grams (g), kilograms (kg) and tonnes. These are known as metric units of mass.

1 kg= 1000 g

1 tonne = 1000 kg

• Ounces and pounds are old units of mass. These are known as imperial units.

• There are 16 ounces in a pound .

• An ounce is roughly equal to 25 grams.

• A pound ( 454 g) is equal to just under half a kilogram (500 g).

MEASURING LENGTH

Length is a measure of how long or wide something is. Rulers and tape measures can

be used to measure length.

• Length is measured in millimetres (rom), centimetres (em), metres (m) or kilometres (km). These are known as metric units of length.

1 cm= l0mm

1m= 100cm

1 km= 1000m

• Miles, feet and inches are old units of length. These are known as imperial units of length.

There are 12 inches in a foot.

An inch is roughly equal to 2.5 centimetres.

A foot is roughly equal to 30 centimetres.

A mile is roughly equal to 1.5 kilometres.

MEASURING CAPACITY OR VOLUME

Capacity or volume is a measure of how much space something takes up. Measuring spoons or measuring jugs can be used to measure capacity.

• Capacity is measured in millilitres (mL) and litres (L). 1L=1000mL

• Pints and gallons are old units of capacity (imperial units). There are 8 pints in a gallon.

A pint is equal to just over half a litre. A gallon is roughly equal to 4.5 litres.

In the laboratory, volume of liquids or solutions can be measured by graduated cylinder, burette, pipette, volumetric or measuring flask etc. A volumetric flask is used to prepare a known volume of a solution. These measuring devices are shown in Fig. 2.6.

To read a scale, first work out how much each mark or division on the scale represents

There is one mark between each 100 mL. So each 100 mL is divided into 2 parts.

100 / 2=50.

So each mark must represent 50 mL

MEASURING TIME

Time is measured by simple clock, digital dock, etc. (Figs. 2.8 (a), 2.8 (b))

• AM (am) is morning time (all times between 12 midnight and 12 midday).

• PM (pm) is afternoon and evening time (all times between 12 midday and 12 midnight).

• An analogue or a 12 hour clock is shown in Fig. 2.8 (a). An analogue clock is one with a face and hands. It is showing the time . twenty past five (Fig. 2.8 (a)).

If it were twenty past five in the morning, it would be. written as 5:20 am (Fig. 2.8 (b)). : If it were twenty past five in the afternoon, it would be written as 5:20 pm (Fig. 2.8 (b)).

Means it is twenty past five in the morning

Means it is twenty past five in the evening

• Nowadays we have digital or 24 hour clocks. They are also showing the time twenty past five.
• The 24 hour time is the same as the analogue time in the morning (except for the 0 at the  beginning for numbers under 10). So 8:45 am becomes 08:45.
• But in the afternoon, you need to add 12 to convert an analogue time to a 24 hour time. So 8:45   pm becomes 20:45.
• Midday on a 24 hour clock is shown as 12:00.
• Midnight on a 24 hour clock is shown as 00:00.

Units of Time

1 minute = 60 seconds

1 hour = 60 minutes

1 day = 24 hours

1 week = 7 days

1 fortnight= 14 days

1 year= 12 months= 52 weeks = 365 days

1 leap year = 366 days

Remember “30 days has September, April, June and November. All the rest have 31. Except for February alone, which has 28 days clear but 29 each leap year”.

MEASURING TEMPERATURE

The thermometer is used to measure temperature.

There are three common scales to measure temperature (degree Celsius), °F (degree Fahrenheit) and K (kelvin).

Here, K is the SI unit. The thermometers based on these scales are shown in Fig. 2.9. Generally,  the thermometer with celsius scale are calibrated from 0° to 100° where these two temperatures are the freezing point and the boiling point of water respectively. The fahrenheit scale is  represented between 32° to 212°.

The temperatures on two scales are related to each other by the following relationship:

OF= 9 /5(OC) + 320

The kelvin scale is related to celsius scale as follows:

K = o C + 273.15

It is worthwhile to note that the temperature below 0Oc (i.e., negative values) are possible in celsius scale but in kelvin scale, negative temperature is not possible.

ACCURACY AND PRECISION

Accuracy

Accuracy is how close a measured value is to the actual (true) value. Accuracy describes how

close an approximation is to a correct answer. Thus, accuracy “is the measure of the difference Between the experimental value or mean value of a set of measurements and the true value.”

Accuracy = Mean value – True value

Smaller is the difference between the mean value and the true value more is the accuracy.

For example, suppose your math textbook tells you that the value of pi (n) is 3.14. You do a careful measurement by drawing a circle and measuring the circumference and diameter, and then you divide the circumference by the diameter to get a value for pi (1t) of 3.16.

The accuracy of your answer is how much it differs from the accepted value.

In this case, the accuracy is 3.16- 3.14 = 0.02.

Precision

Precision is how close the measured values are to each other. Precision describes how many digits we use to approximate a particular value. In simple words, it is the difference between the measured value and the arithmetic mean value for a series of measurements, i.e.,

Precision = Individual value – Arithmetic mean value

The idea is illustrated by the following examples:

(i) Target shot by arrows. If you shoot a quiver of arrows at a target, several outcomes are possible.

• If all of the arrows that you shoot go straight to the bull’ s eye, then your aim is both accurate and precise.

• If, however, the arrows cluster in an area immediately to the right of the hull’s eye, your aim is precise, but not accurate.

• If your aim is bad and the arrows hit positions scattered all over the target, then your aim is neither accurate nor precise.

(ii) The value of π (pi) is 2217. Suppose the value is written as 3.1417 and 3.1392838. Looking at the value, the second number has higher precision, but it would appear that the first is more accurate. (Actual value is 3.142857143.)

(iii) Measurement of a pencil on two different scales. How long is the pencil? The best you can say looking at the scale I, is ‘about 9 centimeters’. You might guess and say ‘about 9.5 centimeters’, but the decimal place is just a guess. Because the smallest unit on the ruler you are using is one centimeter, the precision of your measurement is to the nearest centimeter.

Now look at the scale-II. Here we are using a different ruler to measure the pencil.

How long is the pencil? The best you can say is ‘about 9.5 centimeters’. Again, you might guess and say ‘about 9.51 centimeters, but the second decimal place is just a guess. Because the smallest unit on the ruler you are now using is one millimeter (one tenth of a centimeter), the precision of your measurement is to the nearest millimeter, or tenth of a centimeter.

This second measurement is more precise, because you used a smaller unit to measure with.

UNCERTAINTY IN MEASUREMENT

The study of chemistry involves both theoretical as well as practical aspect which further deals with qualitative and quantitative measurements. While dealing with calculations the meaningful way is to handle the numbers conveniently and present the data realistically with certainty to the extent possible.

SIGNIFICANT FIGURES

Experimental measurements have some uncertainty associated with them. However, one would always like the results to be precise and accurate. These aspects depend on the accuracy of measuring device and the skill of the operator.

A convenient method of expressing the uncertainty in measurement is to express it in terms of significant figures. In this method, it is assumed that all the digits are known with certainty except the last digit which is uncertain to the extent of± 1 in that decimal place. Thus, a measured quantity is expressed in terms of such a number which includes all digits which are certain and a last digit which is uncertain. The total number of digits in the number of significant figures.

The number of significant figures in a measurement is the number of figures that are known with certainty plus one that is uncertain, beginning with the first non-zero digit.

In order to determine the significant figures in a measured quantity the following rules should be applied.

1. All non-zero digits are significant.

For example, 165 cm has three significant figures; 0.165 has also three significant figures. Similarly, 2006 has four significant figures, 9.05 has three significant figures, etc.

2. Zeros to the left of the first non-zero digit in the number are not significant.

For example, 0.005 g has only one significant figure, 0.026 g has two significant figures.

3. Zeros between non-zero digits are significant.

For example, 2.05 g has three significant figures.

4. Zeros to the right of the decimal point are significant.

For example, 5.00 g, 0.050 g, 0.5000 g have three, two and four significant figures respectively.

5. If a number ends in zeros that are not to the right of a decimal, the zeros may or may not be significant.

For example, 1500 g may have two, three or four significant figures.

The ambiguity in the last point can be removed by expressing the number in scientific notation.

In scientific notation the number is written in the standard exponential form as N x 10n.

N = a number with a single non-zero digit to the left of the decimal point.

n = some integer.

For example, a mass of 1500 g can be expressed in scientific notation in the following forms depending upon whether it has two, three or four significant figures

1.5 x 103 g     (Two significant figures)

1.50 x 103 g   (Three significant figures)

1.500 x 103 g  (Four significant figures)

In these expressions all the zeros to the right of the decimal point are significant. The exponential notation is an excellent way of expressing significant figures in very large or very small  measurements. For example, Avogadro’s constant is expressed as 6.022 x 1023 mol-1 and Planck’s constant as 6.62 x w-34 Js.

Lastly, it may be emphasized that exact integral numbers, such as number of pencils in a dozen pencils or number of grams in a kilogram do not have any uncertainty associated with them and as such these numbers have an infinite number of significant figures.

CALCULATIONS WITH SIGNIFICANT FIGURES

During quantitative studies, the scientists have to do calculations with numbers used for various measured physical quantities. These numbers generally, have different number of significant digits depending upon the accuracy with which a particular measurement is made. While carrying out calculations with these numbers, the rule used is that the accuracy of the final result is limited to the least accurate measurement. In other words, final result cannot be more accurate than the least accurate number involved in the calculation.

In order to understand this let us suppose that in a simple experiment two samples A and B are weighed by balances of different accuracy. Let the weight of sample A be 2.3 g and that of sample B be 14 g. The first weighing has 2 as the certain digit and 3 as the doubtful digit. The second weighing has 1 as the certain digit and 4 as the doubtful digit. If the samples A and B are mixed, the final weight would be expressed as 2.3 + 14 = 16 and not as 16.3. In 16.3 the digit at

unit place is certain whereas it is not so in case of weight of sample B. Thus, if the result is expressed as 16.3, the final result would be more accurate than the one of the measurements involved in the calculation, which is not possible.

SOLVED EXAMPLES

1 Example 2.1 . Express the following numbers to four significant figures

(i) 5.607892                     (ii) 32.392800

(iii) 1.78986 X 103          (iv) 0.007837.

Solution. (i) As the fifth digit 8 is greater than 5, therefore the result will be expressed as 5.608.

(ii) It will be expressed as 32.39. The digit 2 is dropped and since it is less than 5 the figure is not rounded off to next number.

(iii) The fifth digit being greater than 5, therefore the result is expressed as 1.790.

(iv) The four digits to be retained are 7, 8, 3 and 7 therefore the number is expressed as 7.837 x 10-3. The exponential term does not add to significant figures.

Example 2.2 Express the following numbers to three significant figures:

(1) 6.022 x 5.359         (ii) 5.359

(iii) 0.04597                 (iv) 34.216.

Solution. (i) The last digit to be retained is 2 and the digit to be dropped is 2 which is less than 5. The result will be expressed as 6.02 x1023.

(ii) In this case the last digit to be retained is 5 and the digit to  be dropped is 9, which is greater than 5. Hence, the last digit to be retained is increased by one. The number will be written as 5.36.

(iii) The three digits to be retained in this case are 4, 5 and 9. The digit 7 is to be dropped which is greater than 5. Hence, the last digit to be retained will therefore be increased by one. The number will be rounded off to three significant digits as 0.0460.

(iv) In this case the digits 1 and 6 will be dropped. Since, the digit following the last digit to be retained is 1, the last digit will be kept unchanged and the number is written.