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No, drinking 1 liter of water will not make you weigh 1 kg heavier. The weight of the water you drink is already included in your body weight before you drink it. Weight gain or loss occurs when you consume more or less calories than your body uses, not from simply drinking water.
mg is milligram, a measure of weight. Liter is a measure of volume. A liter of WATER would weigh 1 million milligrams- but a liter of alcohol would weigh about 700,000 mg- since alcohol is lighter than water. A liter of mercury would weigh MUCH more, since it is heavier than water. So, to answer your question, we would need to know a liter of WHAT?
A liter of water weighs approximately 33.8 ounces.
It is inappropriate to be talking about "lighter" or heavier" with regard to liters and centiliters (or milliliters or deciliters, etc.) Why? Because the liter (and its fractional parts) is a unit of volume or capacity, not weight. You may say that a liter is 100 times greater in volume than a centiliter. You could say that a liter is ten times greater in volume than a deciliter. You could also say that a deciliter is ten times greater in volume than a centiliter. But one is not necessarily heavier or lighter than another. Unless, of course, you have a liter and centiliter of the same substance! A liter of water, for example, will be 100 times heavier than a centiliter of water -- as long as both are at identical temperatures. (Water at 4 degrees Celsius is denser than water at higher and lower temperatures, so a liter of water at 4 degrees C will weigh more than 100 times a centiliter of water at higher or lower temps.)
Yes, if you drink 1 liter of water, you will temporarily weigh about 1 kilogram more since 1 liter of water has a mass of approximately 1 kilogram. However, this weight is not permanent; your body will eventually process and eliminate the excess water through urine and sweat. Thus, while you may see a short-term increase on the scale, it does not reflect a gain in body fat or overall mass.
One liter of heavy water weighs slightly more than one liter of regular water because heavy water contains deuterium, a heavier isotope of hydrogen. The exact weight can vary, but on average, one liter of heavy water weighs about 1.107 kg.